Solve

Solve

Find all solutions to an equation, inequality, or a system of equations and/or inequalities.

Key Formula

\text{If } f(x) = 0, \text{ then solve for } x \text{ such that the equation is satisfied.}

Where:

$f(x)$ = An expression involving the unknown variable x
$x$ = The unknown variable whose value(s) you are trying to find
$0$ = The value the expression equals (after rearranging, equations can often be written in this form)

Worked Example

Problem: Solve the equation 3x + 7 = 22.

Step 1: Subtract 7 from both sides to begin isolating x.

3x + 7 - 7 = 22 - 7

Step 2: Simplify both sides.

3x = 15

Step 3: Divide both sides by 3.

x = \frac{15}{3} = 5

Step 4: Check by substituting x = 5 back into the original equation: 3(5) + 7 = 15 + 7 = 22. ✓

3(5) + 7 = 22 \quad \checkmark

Answer: x = 5

Another Example

This example differs from the first because a quadratic equation can have more than one solution. Solving means finding ALL values that satisfy the equation, not just one.

Problem: Solve the quadratic equation x² − 5x + 6 = 0.

Step 1: Factor the quadratic expression. Look for two numbers that multiply to 6 and add to −5.

x^2 - 5x + 6 = (x - 2)(x - 3)

Step 2: Set each factor equal to zero (zero product property).

x - 2 = 0 \quad \text{or} \quad x - 3 = 0

Step 3: Solve each simple equation.

x = 2 \quad \text{or} \quad x = 3

Step 4: Check both solutions in the original equation. For x = 2: 4 − 10 + 6 = 0 ✓. For x = 3: 9 − 15 + 6 = 0 ✓.

2^2 - 5(2) + 6 = 0 \;\checkmark \qquad 3^2 - 5(3) + 6 = 0 \;\checkmark

Answer: x = 2 or x = 3

Frequently Asked Questions

What does it mean to solve an equation?

To solve an equation means to find every value of the unknown variable that makes the equation a true statement. You perform valid algebraic operations on both sides—such as adding, subtracting, multiplying, or dividing—to isolate the variable. The result is the solution or solution set.

What is the difference between solving and simplifying?

Solving finds the value(s) of a variable that satisfy an equation or inequality—you end with something like x = 5. Simplifying means rewriting an expression in a more compact or equivalent form without finding a specific value. For example, simplifying 2x + 3x gives 5x, but there is no equation to solve because no equals sign is present.

Can an equation have no solution or infinitely many solutions?

Yes. An equation like x + 1 = x + 2 has no solution because no value of x makes it true; this is called a contradiction. An equation like 2(x + 3) = 2x + 6 is true for every value of x; this is called an identity, and the solution set is all real numbers.

Solve vs. Simplify

	Solve	Simplify
Definition	Find the value(s) of the variable that make an equation or inequality true	Rewrite an expression in a simpler or more compact equivalent form
Requires an equals/inequality sign?	Yes — you need an equation or inequality	No — you work with an expression
Result	A specific value or set of values (e.g., x = 5)	A rewritten expression (e.g., 5x instead of 2x + 3x)
Example	Solve 2x = 10 → x = 5	Simplify 2x + 3x → 5x

Why It Matters

Solving equations and inequalities is the central skill in algebra and appears in virtually every math course from pre-algebra through calculus. Standardized tests (SAT, ACT, GCSEs) heavily test your ability to solve linear, quadratic, and systems of equations. Beyond school, solving is the mathematical foundation for answering real-world questions—calculating distances, budgets, rates, and scientific measurements all require finding unknown values.

Common Mistakes

Mistake: Forgetting to find ALL solutions. For example, when solving x² = 9, students often write only x = 3 and miss x = −3.

Correction: Always consider every possible solution. For quadratics and higher-degree equations, expect multiple solutions. Take the positive and negative square root when undoing a square.

Mistake: Performing an operation on one side of the equation but not the other.

Correction: Whatever you do to one side of an equation, you must do exactly the same to the other side. This preserves the equality. For instance, if you subtract 4 from the left side, subtract 4 from the right side too.

Related Terms

Solution — The value(s) found when you solve
Equation — A statement of equality you solve
Inequality — A comparison statement you can solve
Simultaneous Equations — Multiple equations solved together for shared unknowns
System of Inequalities — Multiple inequalities solved together
Simplify — Rewriting expressions, distinct from solving
Variable — The unknown quantity you solve for
Zero Product Property — Key property used when solving factored equations