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Equations

From simple one-step equations to complex systems — learn to solve them all with confidence.

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What Is an Equation?

An equation is a mathematical statement that two expressions are equal, connected by the "=" sign. Solving an equation means finding all values of the variable(s) that make the statement true.

3x + 5 = 20

The fundamental principle of equation solving: whatever you do to one side, you must do to the other. This preserves the equality while isolating the unknown.

Equations are the backbone of all mathematics. They appear in every branch — from geometric relationships to rate-of-change problems in calculus to statistical hypothesis tests.

Linear Equations

A linear equation in one variable has the form ax + b = c, where the variable x appears only to the first power. The graph of a linear equation in two variables is always a straight line (hence the name).

Solving One-Variable Linear Equations

Example: Solve 4x − 7 = 13

Step 1: Add 7 to both sides: 4x = 20

Step 2: Divide both sides by 4: x = 5

Check: 4(5) − 7 = 20 − 7 = 13 ✓

Equations with Variables on Both Sides

Example: Solve 5x + 3 = 2x + 18

Step 1: Subtract 2x from both sides: 3x + 3 = 18

Step 2: Subtract 3: 3x = 15

Step 3: Divide by 3: x = 5

Equations with Fractions

When an equation contains fractions, multiply every term by the least common denominator (LCD) to clear the fractions first.

Example: Solve x/3 + x/4 = 7

LCD = 12. Multiply every term by 12:

4x + 3x = 84

7x = 84 → x = 12

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Linear equations connect directly to linear functions, whose graphs are straight lines with slope m and y-intercept b in the form y = mx + b.

Quadratic Equations

A quadratic equation has the standard form ax² + bx + c = 0 (where a ≠ 0). These equations can have 0, 1, or 2 real solutions.

Method 1: Factoring

If you can factor the quadratic, set each factor equal to zero (see the Polynomials & Factoring page for more techniques).

Example: Solve x² − 5x + 6 = 0

Factor: (x − 2)(x − 3) = 0

x − 2 = 0 → x = 2   or   x − 3 = 0 → x = 3

Method 2: The Quadratic Formula

x = (−b ± √(b² − 4ac)) / (2a)

The expression Δ = b² − 4ac is called the discriminant. It tells you the nature of the solutions:

  • Δ > 0: Two distinct real solutions
  • Δ = 0: One repeated real solution
  • Δ < 0: Two complex conjugate solutions (see complex numbers)

Example: Solve 2x² + 3x − 5 = 0

a = 2, b = 3, c = −5

Δ = 9 − 4(2)(−5) = 9 + 40 = 49

x = (−3 ± 7) / 4

x = 1   or   x = −5/2

Method 3: Completing the Square

This technique rewrites ax² + bx + c as a(x − h)² + k, revealing the vertex of the parabola.

Example: Solve x² + 6x + 2 = 0

x² + 6x = −2

x² + 6x + 9 = −2 + 9 (add (6/2)² = 9 to both sides)

(x + 3)² = 7

x = −3 ± √7

Quadratic equations appear everywhere: in optimization problems in calculus, area calculations in geometry, projectile motion in physics, and regression analysis in statistics.

Systems of Equations

A system of equations is a set of two or more equations with the same variables. The solution is the set of values that satisfies all equations simultaneously.

Method 1: Substitution

Example: Solve the system

y = 2x + 1

3x + y = 11

Substitute: 3x + (2x + 1) = 11 → 5x = 10 → x = 2, y = 5

Method 2: Elimination

Example: Solve the system

2x + 3y = 12

4x − 3y = 6

Add the equations: 6x = 18 → x = 3, then y = 2

Method 3: Matrices (Cramer's Rule)

For larger systems, matrix methods are far more efficient. See Linear Algebra for Gaussian elimination, and the formula sheet for Cramer's Rule.

Systems of equations connect algebra to linear algebra, where Ax = b is the central problem. In higher dimensions, you can't solve systems by hand — you need matrix methods.

Absolute Value Equations

The absolute value |x| gives the distance of x from zero. To solve |expression| = k (where k ≥ 0), split into two cases:

|ax + b| = k   ⟹   ax + b = k   or   ax + b = −k

Example: Solve |2x − 5| = 9

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Case 1: 2x − 5 = 9 → x = 7

Case 2: 2x − 5 = −9 → x = −2

Solution: x = 7 or x = −2

Radical Equations

A radical equation contains a variable inside a radical (√). Isolate the radical and square both sides — but always check for extraneous solutions!

Example: Solve √(x + 3) = x − 1

Square both sides: x + 3 = x² − 2x + 1

Rearrange: x² − 3x − 2 = 0

Factor/quadratic formula: x = (3 ± √17)/2

Check both in the original equation — reject any that produce a false statement.

Radical equations will reappear in trigonometry (half-angle formulas) and integration (u-substitution with radicals).

Related Topics

Polynomials & Factoring Factor and simplify polynomial expressions to solve equations faster. Functions & Graphs Understand how equations become functions and their graphical representations. Vectors & Matrices Solve large systems of equations using matrix algebra. Derivatives Find where functions reach maxima/minima — a generalization of "solve f'(x) = 0".