From simple one-step equations to complex systems — learn to solve them all with confidence.
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.
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.
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).
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 ✓
Step 1: Subtract 2x from both sides: 3x + 3 = 18
Step 2: Subtract 3: 3x = 15
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 3: Divide by 3: x = 5
When an equation contains fractions, multiply every term by the least common denominator (LCD) to clear the fractions first.
LCD = 12. Multiply every term by 12:
4x + 3x = 84
7x = 84 → x = 12
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.
A quadratic equation has the standard form ax² + bx + c = 0 (where a ≠ 0). These equations can have 0, 1, or 2 real solutions.
If you can factor the quadratic, set each factor equal to zero (see the Polynomials & Factoring page for more techniques).
Factor: (x − 2)(x − 3) = 0
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 − 2 = 0 → x = 2 or x − 3 = 0 → x = 3
The expression Δ = b² − 4ac is called the discriminant. It tells you the nature of the solutions:
a = 2, b = 3, c = −5
Δ = 9 − 4(2)(−5) = 9 + 40 = 49
x = (−3 ± 7) / 4
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 = 1 or x = −5/2
This technique rewrites ax² + bx + c as a(x − h)² + k, revealing the vertex of the parabola.
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² + 6x = −2
x² + 6x + 9 = −2 + 9 (add (6/2)² = 9 to both sides)
(x + 3)² = 7
x = −3 ± √7
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 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.
y = 2x + 1
3x + y = 11
Substitute: 3x + (2x + 1) = 11 → 5x = 10 → x = 2, y = 5
2x + 3y = 12
4x − 3y = 6
Add the equations: 6x = 18 → x = 3, then y = 2
For larger systems, matrix methods are far more efficient. See Linear Algebra for Gaussian elimination, and the formula sheet for Cramer's Rule.
The absolute value |x| gives the distance of x from zero. To solve |expression| = k (where k ≥ 0), split into two cases:
Case 1: 2x − 5 = 9 → x = 7
Case 2: 2x − 5 = −9 → x = −2
Solution: x = 7 or x = −2
A radical equation contains a variable inside a radical (√). Isolate the radical and square both sides — but always check for extraneous solutions!
Square both sides: x + 3 = x² − 2x + 1
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: x² − 3x − 2 = 0
Factor/quadratic formula: x = (3 ± √17)/2
Check both in the original equation — reject any that produce a false statement.