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Mathematical Foundations⌄

Number Systems and Quantities⌄

Integer, fraction, irrational and real numbers⌄

Comparison of numbers on the number axis to absolute values⌄

Four Operations on Signed Numbers⌄

Order of operations and properties of operations⌄

Units, Volumes, Ratios, Rates, Proportions and Percentages⌄

Complex Numbers, Vectors, and Matrices⌄

Representation of complex numbers and four operations⌄

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Plural conjugate and plural quotient⌄

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Size, direction, and component of the vector⌄

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Addition and subtraction of vectors and multiplication⌄

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Representations, operations, determinants and inverse matrices of matrices⌄

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Solve equations using augmented matrices⌄

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Mathematical Language and Structure⌄

Convert text to mathematical expressions⌄

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Variables, constants, coefficients, terms and factors⌄

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Intervals, Sets, and Inequality Representations⌄

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Variables, constraints, and fields in formulas⌄

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Checks if a number satisfies an equation or model⌄

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Algebra⌄

Expressions and Equivalent Forms⌄

Combine like terms.⌄

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Use the Distributive Property.⌄

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Addition, subtraction and multiplication of polyn⌄

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Special product formula⌄

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Zero, negative, and rational exponents⌄

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Rewrite Expression to Equivalent Form⌄

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Linear Equations in One Variable⌄

Solve one-step and multi-step one-step equations⌄

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Solve one equation with multiple variable terms⌄

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Judgement without solution and infinite solution⌄

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Unletter Formula⌄

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Solve absolute-value equations⌄

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Represent the solution set on the number axis⌄

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Linear Inequalities in One Variable⌄

Solve one-step, multi-step and compound inequalities⌄

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Solve absolute-value inequalities⌄

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Systems of Linear Equations⌄

Setting up equations from real-world situations⌄

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Solving equations with image method⌄

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Solve equations by substitution⌄

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Solve the system of equations with the elimination method⌄

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Determine if the system of equations has one, no, or infinitely many solutions⌄

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Solve Ternary Primary Equations⌄

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Solve mixing, rate, cost, and nodal problems with equations⌄

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Systems of Linear Inequalities and Feasible Regions⌄

Illustration of feasible domains for linear inequality groups⌄

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Polynomials and Factoring⌄

Determine the number of polynomials, first term coefficients, and terminal behavior⌄

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Extract Maximum Equations⌄

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Factorization Squared Variance⌄

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Factorization Quadratic Trinomial⌄

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Fully factorized⌄

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binomial theorem⌄

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Polynomial Division⌄

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Remainder Theorem, Factor Theorem and Rational Root Theorem⌄

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Rational and Radical Expressions⌄

Determine rational constraints⌄

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Reduction Rational Equation⌄

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Rational multiplication and division⌄

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Rational addition and subtraction⌄

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Simplify complex rational expressions⌄

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Simplified radical⌄

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Radical addition, subtraction, multiplication and division⌄

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Denominator rationalization⌄

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Solve the root equation and exclude additional roots⌄

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Functions and Modeling⌄

Function Concepts and Representations⌄

Determine if the relationship is a function⌄

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Evaluate using function notation⌄

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Define domains and ranges from rules, tables, images, or contexts⌄

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Distinguish between independent and dependent variables⌄

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Identify intercepts, zeros, extrema, symmetry, and end behavior⌄

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Functions Comparing Equations, Tables, Images, and Text Representations⌄

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Segmented Functions and Continuity⌄

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Linear Functions and Models⌄

Calculate and interpret rate of change⌄

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Calculate the slope from two points, a table or an image⌄

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Writing oblique, dot oblique and general⌄

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Write parallel and perpendicular equations⌄

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Draw linear functions based on equations or tables⌄

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Modeling with Linear Functions⌄

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Explain the actual meaning of slope and intercept⌄

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Linear Regression, Correlation, Causality and Residuals⌄

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Absolute Value, Radical, and Variation Models⌄

Draw Absolute Value Function⌄

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Translation, Stretching and Reflection Absolute Value Functions⌄

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Draw square root functions and cube root functions⌄

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Direct, Counter, and Combined Changes⌄

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Modeling with Roots or Change Functions⌄

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Quadratic Functions and Models⌄

Identify the vertex, axis of symmetry, intercepts, maximum, and minimum⌄

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Draw vertex and general quadratic functions⌄

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Converting between general, vertex, and factoring expressions⌄

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Modeling with Quadratic Functions⌄

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Compare linear, exponential and quadratic models⌄

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Image Method Analysis Nonlinear Intersection Problem⌄

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Solving quadratic equations with image method⌄

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Solve quadratic equations with open square root method⌄

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Solve quadratic equations with matching methods⌄

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Solve quadratic equations with quadratic equations⌄

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Polynomial and Rational Functions⌄

Draw polynomial functions based on zero, multiplicity and end behavior⌄

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Determine increase/decrease intervals and inflection points⌄

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Determine the zero point according to the factorization formula⌄

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Determine the asymptotes of rational functions⌄

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Draw rational functions based on constraints and asymptotes⌄

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Solve polynomial equations and rational equations⌄

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Fit data with polynomials or rational functions⌄

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Function Operations, Composition, and Inverse Functions⌄

Function addition, subtraction, multiplication and division⌄

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Functional Composition⌄

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Determine if the function corresponds one-to-one⌄

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Inverse function relation or inverse function⌄

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Verify that two functions are inverses⌄

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Translate, reflect, stretch, and compress functions⌄

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Exponential and Logarithmic Functions⌄

Exponential algorithm⌄

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Exponential growth and decay image⌄

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Write exponential functions based on tables, images or situations⌄

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Simple interest, compounding and sequential compounding⌄

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Exponential and logarithmic swaps⌄

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Logarithmic product, quotient, and power laws⌄

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Draw a logarithmic function⌄

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Solve exponential and logarithmic equations⌄

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Exponential and Logarithmic Models⌄

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Geometry⌄

Geometric Foundations, Logic, and Proof⌄

Points, lines, faces, segments, rays and angles⌄

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Definitions, Axioms, Assumptions and Theorems⌄

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Conditional propositions, inverse propositions, no propositions, inverse-no propositions, and biconditional propositions⌄

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Laws of conjunction, disjunction and logic⌄

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Construct counter-examples⌄

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Double Column Proof, Paragraph Proof, Process Proof and Coordinate Proof⌄

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Prove relationships involving segments, angles, and geometric figures⌄

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Use Disproof⌄

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Lines, Angles, and Coordinate Geometry⌄

Midpoint and two-point distance⌄

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Linear slope between two points⌄

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Linear equation in coordinate plane⌄

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Parallel to perpendicular⌄

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Prove line parallelism using angular relationships⌄

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Polygon inner and outer corners and⌄

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Prove geometry using coordinate method⌄

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Transformations, Symmetry, and Congruence⌄

About Line or Axis Reflection⌄

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Pan using vector or coordinate rules⌄

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Rotate around dots⌄

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Perform a similarity transformation and determine the scale factor⌄

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Composite Transform⌄

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Prove full equivalence with rigid transformations⌄

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Line symmetry, point symmetry and rotational symmetry⌄

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Prove triangle congruence using SSS⌄

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Prove triangle congruence using SAS⌄

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Prove triangle congruence using ASA or AAS⌄

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Prove right triangle congruence using HL⌄

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Triangle Relationships and Similarity⌄

Sort triangles by edges and corners⌄

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Triangle internal angle and theorem⌄

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Isosceles and equilateral triangles theorem⌄

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Midline, High, Angular Split Line and Vertical Split Line⌄

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Triangle Inequality⌄

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Prove polygonal similarity with telescopic transformations⌄

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Prove triangle similarity using AA, SSS or SAS⌄

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Triangle Segment Scale Relationship⌄

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Use Similarity to Solve Proportional and Indirect Measurement Problems⌄

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Quadrilaterals and Polygons⌄

Quadrilateral by nature⌄

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Properties of Parallelograms⌄

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Prove that a quadrilateral is a parallelogram⌄

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Properties of rectangles, diamonds and squares⌄

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Trapezoidal and kite-shaped properties⌄

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Polygon Perimeter and Area⌄

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Use Edge, Angle, and Diagonal Properties to Find Unknown Quantities⌄

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Circles and Loci⌄

Corners, arcs, strings, tangents and tangents⌄

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Arc Measurement and Arc Length⌄

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Inner Junction Angle and Pairwise Arc⌄

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Angles formed by strings, tangents and tangents⌄

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Length of tangent, chord and tangent segments⌄

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Circular equations in the coordinate plane⌄

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Geometric trajectory⌄

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Plot parallel lines, vertical bisections, and angular bisections⌄

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Measurement and Three-Dimensional Geometry⌄

Circle circumference, area and sector⌄

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Prisms, Pyramids, Cylinders, Cone and Spheres⌄

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Surface area of polyhedra, cylinders, cones and spheres⌄

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Volume of polyhedra, cylinders, cones and spheres⌄

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Sections and Unfolding⌄

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Area Ratio and Volume Ratio of Similar Graphics⌄

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Geometric measurements in real-world scenarios⌄

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Right Triangles and Trigonometric Applications⌄

Pythagorean theorem⌄

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Sine, Cosine and Tangent in Right Triangle⌄

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Use trigonometric ratio to calculate unknown edges⌄

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Unknown angle using inverse trigonometric function⌄

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Right Triangle Application Question⌄

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Sine Theorem⌄

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Cosine theorem⌄

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Ambiguity of SSA⌄

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Use the triangle information to calculate the area of a non-right triangle⌄

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Trigonometry and Precalculus⌄

Angles, Radians, and the Unit Circle⌄

Swap angles with radians⌄

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Standard Position Angle and Quadrant⌄

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Determine sine and cosine by unit circle⌄

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Exact triangular values for special angles⌄

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Reference Angle and Quadrant Symbols⌄

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Calculate six trigonometric function values from coordinates or unit circles⌄

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Trigonometric Function Graphs⌄

Inverse trigonometric function⌄

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Trigonometric function domains and ranges⌄

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Sine, cosine, tangent and reciprocal function images⌄

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Amplitude, period, phase shift, midline and vertical shift⌄

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Write sine, cosine equations according to the image or situation⌄

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Inverse trigonometric image and evaluation⌄

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Remainder Function Relationship⌄

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Trigonometric Identities⌄

Reciprocal Identities, Quotient Identities and Stock Hook Identities⌄

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Verify a trigonometric identity⌄

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Sum difference formula⌄

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Double angle formula⌄

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Half-width and power-down formulas⌄

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Sum differential product and product and formula⌄

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Trigonometric Equations and Periodic Models⌄

Solve the first trigonometric equation⌄

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Factorization to solve trigonometric equations⌄

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Quadratic formula for solving trigonometric equations⌄

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Solving trigonometric equations by commutation⌄

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Solve trigonometric equations within a specified interval⌄

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Cycle Modeling with Trigonometric Functions⌄

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Explain the amplitude, period and phase in the periodic model⌄

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Analytic Geometry, Sequences, and Series⌄

Swap rectangular coordinates with polar coordinates⌄

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Image and nature of cone curve⌄

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Center, vertex, focal point, axis and asymptote of cone curve⌄

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Equations with conical curves⌄

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Array and Integer Index Functions⌄

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Sigma Summation Symbol⌄

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Equal Difference Series and Equal Difference Series⌄

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Equal Ratio Series and Equal Ratio Series⌄

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Explicit and Recursive Definition of Sequences⌄

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Convergence and Sum of Infinity Ratio Series⌄

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Calculus Interface⌄

Understand limits from tables, images, or functions⌄

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Estimated instantaneous rate of change⌄

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Understand derivative as rate of change or tangent slope⌄

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Understand definite integral as cumulative change or area⌄

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Connect graphical, numerical, and algebraic representations of change⌄

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Statistics and Probability⌄

Data Collection and Study Design⌄

Population vs. Sample⌄

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Observational studies, experiments and investigations⌄

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Explain variables, response variables, categorical variables and quantitative variables⌄

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Deviations and confounding variables⌄

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Conditions for the Establishment of Causal Conclusions⌄

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Frequency tables⌄

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Data Displays and Descriptive Statistics⌄

Dots, histograms and box plots⌄

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Bar Charts and Categorized Data Charts⌄

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Mean, Median, Multiplier, Quartile and Percentile⌄

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Variance, Quartile Range, Variance and Standard Deviation⌄

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Compare distribution with center, degree of discreteness, shape and outliers⌄

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Group data statistics⌄

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Cumulative Frequency and Percentile⌄

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Bivariate Data, Correlation, and Regression⌄

Interpreting the Scatterplot⌄

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Describe the direction, shape, intensity and outliers of the correlation⌄

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Calculate and interpret correlation⌄

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Establish and interpret linear regression lines⌄

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Building and Interpreting Nonlinear Regression Models⌄

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Differentiate interpolation from extrapolation⌄

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Using the Residual Evaluation Model⌄

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Probability, Counting, and Conditional Probability⌄

Sample space and events⌄

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Empirical and theoretical probabilities⌄

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Complements, merges, and intersections⌄

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Addition rule⌄

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Multiplication rule⌄

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Conditional probability⌄

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Determine event independence⌄

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Tree, Column, and Bidirectional Tables⌄

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Basic Counting Principle⌄

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Arrange & Combine⌄

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Multistage Probability Problem⌄

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Random Variables and Probability Distributions⌄

Random variables and probability distributions⌄

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Expected value⌄

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Binomial probability⌄

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binomial theorem⌄

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Normal distribution and its parameters⌄

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Z-score⌄

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Normal curve probability⌄

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Statistical Inference⌄

Statistics and Parameters⌄

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Sampling Distributions⌄

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confidence interval⌄

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Margin of error⌄

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Original and Alternative Assumptions⌄

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p value⌄

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Statistical vs. Actual Significance⌄

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Modeling and Reasoning⌄

Modeling Process⌄

Determining Quantities, Units, Variables, and Constra⌄

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Choose an equation, inequality, table, graph, or function representation⌄

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Select model type based on structure⌄

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Solve or estimate in the model⌄

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Explain the results back to the real situation⌄

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State model assumptions and limitations⌄

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Mathematical Reasoning and Communication⌄

Use definitions, properties and qualitative interpretations⌄

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Compare different problem-solving methods⌄

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Negating universal propositions with counter-examples⌄

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Discover patterns and validate campaigns⌄

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Steps to Identify Errors in Mathematical Arguments⌄

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Express results with the right symbols, units, and domain⌄

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x + 7 = 19

Math · Foundations

How to solve linear equations with confidence

Build a reliable method for keeping both sides balanced.

✓ steps

Study skills · Methods

How to show work that earns credit

Make your reasoning visible without making it harder to follow.

90 min

Exam prep · Planning

A calm plan for your next math exam

Use short practice cycles to turn revision into recall.

• ─ •

Math · Geometry

Understand the midpoint formula, not just the steps

Connect coordinates, averages, and a visual model.

↺ learn

Study skills · Reflection

Turn common mistakes into a study guide

A simple review habit for finding the next useful practice.

? → x

Math · Problem solving

Translate word problems into equations

Find the quantities, relationship, and one useful unknown.

2, 4, 6, …

Math · Patterns

Find the rule before extending a sequence

Compare consecutive terms to identify a useful relationship.

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Exam prep · Revision

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MATH · LINEAR EQUATIONS

How do I solve a linear equation?

Algebra 1MediumUpdated Sep 16, 2026

Demo access 3 free solutions remaining

Question

Solve 5x - 7 = 18.

Use inverse operations to keep both sides balanced while isolating x.

Step-by-step review

1Add 7 to both sides: 5x = 25.✓ Correct
2Divide both sides by 5: x = 5.✓ Correct

what you got right

Your separation of the constant was clean, and you correctly applied the same operation to both sides. You isolated the variable in the right order and checked the result against the original equation.

Why it happened

The Slip

Remember to apply the inverse operation to the complete side of the equation.

Deep Root Cause & Pedagogical Diagnosis

This step checks whether the variable is isolated before the final answer is recorded.

Targeted fix

checkpoint1 · Separate the constant from the variable.

checkpoint2 · Apply the same operation to both sides.

checkpoint3 · Isolate the variable.

checkpoint4 · Substitute to check the result.

Corrected path

5x − 7 = 18 → 5x = 25 → x = 5. Substitute 5 back into the original equation to confirm the result.

Model to transfer

Trigger Signal

A variable is surrounded by addition or subtraction and multiplication or division.

3-Step Routine

Separate the constant → isolate the variable → substitute to check.

Answer

x = 5

Explanation

Apply inverse operations in the same order to both sides of the equation.

Solution steps

Add 7 to both sides, then divide both sides by 5. Each operation preserves equality.

6-part thinking model

Identify the target, choose the inverse operation, preserve balance, calculate, check, and explain.

Lock it in with practice

Try a new equation using the same model.

Rate this answer

Was this explanation helpful?

Expression	Position of 0	Value	Status
0/1 or 0/9	Numerator	0	Valid rational
7/0 or −4/0	Denominator	Undefined	Not a real number
0/0	Both	Indeterminate	Undefined

Set	Does 0 belong?	Reason
Real numbers	Yes	0 is a point on the real number line.
Rational numbers	Yes	0 = 0/1.
Irrational numbers	No	Rational and irrational sets do not overlap.
Integers	Yes	0 has no fractional part.
Natural numbers	Usually no*	Most school definitions begin at 1.

Feature	π	22/7
Type	Irrational	Rational
Decimal	Non-terminating, non-repeating	Repeating 142857
Exact?	Yes	No, approximation
Difference	22/7 − π ≈ 0.001264

Statement	Verdict	Reason
π can be written as a/b	False	That would make π rational.
22/7 is a rational number	True	22 and 7 are integers and 7 ≠ 0.
π ≈ 3.14	True	The symbol ≈ signals approximation.
π = 22/7	False	The values differ by a non-zero amount.
16π is exact	True	Keep π for an exact area.
22/7 may be used when asked for an estimate	True	It is a convenient rational estimate.

Number or claim	Classification	Reason
3/8	Rational	It is a ratio of integers.
√5	Irrational	It cannot be written as a fraction of integers.
√49	Rational	√49 = 7 = 7/1.
π − π	Rational	The result is 0.
√3 × √3	Rational	The result is 3.
Both rational and irrational	Impossible	ℚ ∩ 𝕀 = ∅.

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Blog

How to solve linear equations with confidence

How to show work that earns credit

A calm plan for your next math exam

Understand the midpoint formula, not just the steps

Turn common mistakes into a study guide

Translate word problems into equations

Find the rule before extending a sequence

Make a revision checklist you will actually use

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How to solve linear equations with confidence

The foundation: balance

A repeatable method

Work through an example

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Understanding linear equations

Start with the basic idea

See it from more than one angle

Work through an example

Conclusion

Citations

How do I solve a linear equation?

Question

You've used your 3 free solutions.

Step-by-step review

what you got right

Why it happened

The Slip

Deep Root Cause & Pedagogical Diagnosis

Targeted fix

Corrected path

Model to transfer

Trigger Signal

3-Step Routine

Answer

Explanation

Solution steps

6-part thinking model

Lock it in with practice

Typical errors

Rate this answer

Typical error

Step-by-step review

Was this explanation helpful?

1Why students think 0 is “neither”

“0 means nothing, so it is not a number.”

“Fractions cannot contain 0.”

“Rational numbers must look like fractions.”

2Why 0 is an integer (ℤ)

3Why 0 is a rational number (ℚ)

4The boundary rule: numerator vs. denominator

5Where 0 fits in the real number system

6Practice check

7Next Steps & Related Lessons

1Why students think π equals 22/7

“My calculator uses 22/7, so they are equal.”

“Both decimals keep going, so both are irrational.”

“Any familiar fraction for π must be π.”

2Formal proof: exact value vs. approximation

3Compare the two numbers

4Boundary checklist

5Practice check

6Next Steps & Related Lessons

1Why students think both labels can apply

“Integer and rational are opposites.”

“Every square root is irrational.”

“An irrational expression always stays irrational.”

2Formal proof: the sets are mutually exclusive

3Set relationship diagram

4Boundary checklist

5Practice check

6Next Steps & Related Lessons