Grade 7 Equations: Solve Linear Equations Step-by-Step (US Common Core)

GRADE 7 MATHEMATICS • LESSON 7

Equations

المعادلات

An equation is a balance. Learn to solve one-step and two-step equations using inverse operations, keep both sides equal, check your answer, and write equations from words — then take 15 exams with full step-by-step solutions.

What is an equation?

An equation is a statement that two things are equal — it always has an equals sign. Think of it as a balance scale: whatever is on the left weighs exactly the same as what is on the right.

To solve an equation means to find the value of the letter that keeps the scale balanced. The golden rule: whatever you do to one side, you must do to the other, so it stays balanced.

x + 3
=
8
Read it: “some number plus 3 equals 8”. Take 3 off both sides and x = 5.

Words you must know

The exact vocabulary of solving equations.

Equation
A number sentence with an equals sign, like x + 3 = 8.
Solve
To find the value of the letter that makes the equation true.
Solution
The value of the letter that works. For x + 3 = 8 the solution is x = 5.
Inverse operation
The opposite operation. + and − are inverses; × and ÷ are inverses.
Balance rule
Do the same thing to both sides so the equation stays equal.
⚖️ Two-Step Equation Solver

Set the numbers in a·x + b = c and watch it get solved step by step.

Working with equations

Your main study reference. Each method is broken into numbered steps, each one explained, with a worked example and a teacher hack.

1) One-step equations: + and −

  1. Look at what is done to x. If a number is added, you will subtract it; if subtracted, you will add it.
  2. Do the inverse to both sides. This keeps the balance and leaves x alone.
  3. Read off the answer and check it.
Example: x + 7 = 12
Subtract 7 from both sides: x = 12 − 7
x = 5   (check: 5 + 7 = 12 ✓)
Hack: the inverse of + is −, and the inverse of − is +. Move the number to the other side and flip its sign.

2) One-step equations: × and ÷

  1. See how x is tied to the number. If x is multiplied, you will divide; if divided, you will multiply.
  2. Do the inverse to both sides.
  3. Check by putting the answer back in.
Example: 4x = 20
Divide both sides by 4: x = 20 ÷ 4
x = 5   (check: 4 × 5 = 20 ✓)
Hack: 4x means “4 times x”, so the inverse is ÷ 4. Always divide by the coefficient.

3) Two-step equations

  1. Undo the + or − first. Move the plain number to the other side.
  2. Then undo the ×. Divide both sides by the coefficient of x.
  3. Check your solution in the original equation.
Example: 3x + 5 = 20
Subtract 5: 3x = 15
Divide by 3: x = 5   (check: 3×5 + 5 = 20 ✓)
Hack: work in reverse order of operations — undo + and − before × and ÷.

4) Writing an equation from words

  1. Name the unknown. Let the number be x.
  2. Translate the sentence into symbols, including the equals sign.
  3. Solve the equation you built.
Example: “3 times a number, plus 4, is 19”
3x + 4 = 19
3x = 15, so x = 5
Hack: the word “is” usually becomes the equals sign.
✅ Solution Checker

Is your value of x correct? Test it in a·x + b = c.

Test Yourself — 15 Exams

Each exam has 10 questions, and every answer comes with a step-by-step solution — even when you get it right.

Key takeaways

1. An equation has an equals sign and behaves like a balance scale.

2. Solve by doing the same inverse operation to both sides.

3. Inverses: + undoes −, × undoes ÷.

4. Two-step: undo + or − first, then undo × or ÷.

5. Always check by substituting your answer back into the original equation.