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Equation

Equation

A mathematical sentence built from expressions using one or more equal signs (=).

 

Examples:

 Math equation example: 1/x = x + 1

The equation 3x minus 4y equals 1, written as: 3x - 4y = 1

log base 2 of (x−1) plus log base 2 of (x+3) equals 5

 

See also

Equation rules, inequality, equation of a line, solution set, verify a solution

Key Formula

ax+b=cax + b = c
Where:
  • aa = The coefficient of the variable (a number multiplying x)
  • xx = The unknown variable you solve for
  • bb = A constant term on the same side as the variable
  • cc = A constant on the other side of the equal sign

Worked Example

Problem: Solve the equation 3x + 7 = 22.
Step 1: Subtract 7 from both sides to isolate the term with x.
3x+7−7=22−73x + 7 - 7 = 22 - 7
Step 2: Simplify both sides.
3x=153x = 15
Step 3: Divide both sides by 3 to solve for x.
3x3=153\frac{3x}{3} = \frac{15}{3}
Step 4: Simplify to find the value of x.
x=5x = 5
Step 5: Verify by substituting x = 5 back into the original equation: 3(5) + 7 = 15 + 7 = 22. ✓
3(5)+7=22✓3(5) + 7 = 22 \quad \checkmark
Answer: x = 5

Another Example

This example involves variables on both sides of the equal sign and requires distributing before solving, showing a more complex one-step progression beyond the basic ax + b = c form.

Problem: Solve the equation 2(x − 4) = x + 6.
Step 1: Distribute the 2 on the left side.
2x−8=x+62x - 8 = x + 6
Step 2: Subtract x from both sides to collect variable terms on one side.
2x−x−8=62x - x - 8 = 6
Step 3: Simplify the left side.
x−8=6x - 8 = 6
Step 4: Add 8 to both sides to isolate x.
x=14x = 14
Step 5: Verify: 2(14 − 4) = 2(10) = 20, and 14 + 6 = 20. Both sides match. ✓
2(14−4)=14+6=20✓2(14 - 4) = 14 + 6 = 20 \quad \checkmark
Answer: x = 14

Frequently Asked Questions

What is the difference between an equation and an expression?
An expression is a combination of numbers, variables, and operations (like 3x + 7) that represents a value but makes no claim about equality. An equation contains an equal sign and states that two expressions have the same value (like 3x + 7 = 22). You solve an equation, but you simplify or evaluate an expression.
What is the difference between an equation and an inequality?
An equation uses an equal sign (=) and states that two expressions are exactly equal. An inequality uses symbols like <, >, ≤, or ≥ and states that one expression is greater than or less than another. Equations typically have a finite number of solutions, while inequalities often have infinitely many solutions represented as a range.
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 can make 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. Most equations you encounter in algebra have exactly one solution or a specific finite set of solutions.

Equation vs. Expression

EquationExpression
DefinitionA statement that two expressions are equal, using an = signA combination of numbers, variables, and operations with no = sign
Contains an equal sign?Yes, alwaysNo, never
Example3x + 7 = 223x + 7
What you do with itSolve it to find the unknown value(s)Simplify it or evaluate it for given values
ResultA solution (e.g., x = 5)A simplified expression or a numerical value

Why It Matters

Equations are the foundation of algebra and appear in nearly every area of mathematics, science, and engineering. From calculating the slope of a line to balancing a chemical reaction, equations let you model real-world relationships and find unknown quantities. Mastering how to set up and solve equations is essential for success in algebra, geometry, physics, and beyond.

Common Mistakes

Mistake: Performing an operation on only one side of the equation.
Correction: Whatever you do to one side, you must do to the other side to keep the equation balanced. If you subtract 7 from the left side, you must also subtract 7 from the right side.
Mistake: Confusing an expression with an equation and trying to 'solve' an expression.
Correction: You can only solve a statement that has an equal sign. If there is no equal sign, you have an expression — you can simplify or evaluate it, but there is nothing to solve.

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