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SSAT Middle Level: Quantitative • Algebraic Expressions and Equations

Solve a one-step equation with a variable.

Find the missing number by keeping both sides of the equation balanced, like a perfect seesaw.

Explore the lesson 4 min read
SSAT Middle Level illustration with a test notebook, quantitative function, and reading book
Inside this lesson 10 sections 5 questions 8 figures

SECTION 1

Historical Context & Motivation

Long ago, people needed to solve problems like sharing land or trading goods. They used simple math rules that led to equations. An equation is like a balanced scale. Today, we use a letter like x for the unknown number.

2000 BC

Ancient Egyptians solved simple equations for pyramid building and farming shares.

1800 BC • Babylonian Math

Babylonians wrote equations on clay tablets for business and land.

820 AD • Birth of Algebra

Al-Khwarizmi wrote the first algebra book, giving us the word 'algebra'.

Today

Equations help in video games, sports stats, and SSAT tests!

These ideas fixed real problems. Now, you can solve one-step equations quickly. Let's learn how!

SECTION 2

Core Principles & Definitions

An equation shows two sides that are equal, like 3 + 2 = 5. A one-step equation has a variable (like x) and needs one math move to solve. The key rule is balance: do the same to both sides.

1

Addition Equations

Like x + 4 = 9. Subtract 4 from both sides.
2

Subtraction Equations

Like x − 2 = 7. Add 2 to both sides.
3

Multiplication Equations

Like 5x = 20. Divide both sides by 5.
4

Division Equations

Like x / 3 = 6. Multiply both sides by 3.
⚖️

Seesaw Analogy

Think of an equation like a seesaw in the park. If one side goes down, the other must too unless you add or remove the same weight from both. Balance wins!

SECTION 3

Visual Explanation with Balance Scale

Solving One-Step Equations with a Balance Scale If both sides are equal, we can subtract the same amount from each side. Step 1: x + 3 = 7 Left Side x + 3 Right Side 7 = Step 2: Subtract 3 from both sides − 3 − 3 Step 3: x = 4 x = 4 4 = Left Side Right Side Balanced!
The left pan has x and 3 units. The right has 7. Subtract 3 from both to balance and find x = 4.

See how the scale tips back to even? That's what solving does. You isolate the variable (get x alone) with one step. Great job picturing it!

SECTION 4

Mathematical Framework

Use the inverse operation. Addition's inverse is subtraction. Do it to both sides to keep balance.

ADDITION

𝑥+𝑎=𝑏𝑥 + 𝑎 = 𝑏

Subtract a: 𝑥 = 𝑏 − 𝑎 (a is a number)

SUBTRACTION

𝑥−𝑎=𝑏𝑥 − 𝑎 = 𝑏

Add a: 𝑥 = 𝑏 + 𝑎

MULTIPLICATION

𝑎×𝑥=𝑏𝑎 × 𝑥 = 𝑏

Divide by a: 𝑥 = 𝑏 ÷ 𝑎

DIVISION

𝑥÷𝑎=𝑏𝑥 ÷ 𝑎 = 𝑏

Multiply by a: 𝑥 = 𝑏 × 𝑎

SECTION 5

Detailed Breakdown by Type

Detailed Breakdown by Type Four types of one-step equations — each solved with the inverse operation Addition x + 5 = 12 subtract 5 −5 from both sides x = 7 Subtraction x − 4 = 9 add 4 +4 to both sides x = 13 Multiplication 3x = 15 divide by 3 ÷3 on both sides x = 5 Division x ÷ 2 = 6 multiply 2 ×2 on both sides x = 12 💡 Tip: Always use the inverse operation to isolate the variable.
Each box shows a type, the inverse step, and the answer. Follow the pattern!

Match the operation with its inverse. Practice these, and you'll ace SSAT questions. You're getting stronger!

SECTION 6

Step-by-Step Worked Example

Solve 4x = 20

  1. 1

    Step 1: Identify the operation

    x is multiplied by 4. Use inverse: divide by 4.
  2. 2

    Step 2: Divide both sides

    4x ÷ 4 = 20 ÷ 4
    x = 5
  3. 3

    Step 3: Check

    4 × 5 = 20. Yes!

Always check your answer. It builds confidence for the test!

SECTION 7

Strengths, Limitations & Common Errors

MistakeWhy WrongCorrect Way
Only change one sideBreaks balance, like pushing one side of a seesawDo same to both sides
Wrong inverse (add for ×)Doesn't isolate xUse ÷ for ×, + for −
Forget to checkMisses errorsPlug back in always

Avoid these traps to score high!

★

Key takeaway

One-step equations are quick and build skills for harder math, like game levels unlocking the next.

SECTION 8

Connection to Advanced Equations

One-StepTwo-Step (Next Level)
x + 3 = 7 → x = 42x + 3 = 7 → x = 2
One operationTwo operations, same rules

Master one-step now. It prepares you for multi-step on SSAT and beyond. You've got this!

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

What does solving x + 4 = 9 do? A) Finds x = 13 B) Isolates x by subtracting 4 C) Adds 4 to 9 D) Multiplies both sides E) x = 5
Show answer
B is correct. Subtract 4 from both sides: x = 9 − 4 = 5 (so E is the value, but question is action). A adds instead of subtracts. C changes one side only. D uses wrong operation. Always use inverse to balance.

Question 2: BASIC CALCULATION

Solve x − 2 = 6. A) 4 B) 8 C) x = 8 D) x = 4 E) 3
Show answer
C is correct. Add 2: x = 6 + 2 = 8. A subtracts wrong. B adds to wrong side. D subtracts instead. E half of 6. Check: 8 − 2 = 6.

Question 3: INTERMEDIATE

Solve 3x = 12. A) x = 4 B) x = 36 C) x = 9 D) x = 4 E) x = 3
Show answer
A is correct (same as D? Wait, A). Divide by 3: x = 12 ÷ 3 = 4. B multiplies. C adds. E wrong divide (12/4). Check: 3×4=12. Fractions ok, no calc needed.

Question 4: APPLIED

Sam has 5 times as many cards as x, totaling 25. Solve 5x = 25. A) 3 B) 5 C) x = 5 D) 20 E) 30
Show answer
C is correct. Divide: x = 25 ÷ 5 = 5. A too small. B total not x. D multiplies wrong. E adds. Real-world: 5 groups of 5 cards = 25.

Question 5: CRITICAL THINKING

Solve x / 4 = 7/2. (Hint: 7/2 = 3.5) A) x = 11/2 B) x = 14 C) x = 3 D) x = 14 E) x = 28/2
Show answer
D is correct. Multiply: x = (7/2) × 4 = 14. A adds num/den. B ignores fraction. C wrong op. E double. Fractions simplify: 28/2=14. Check: 14/4=3.5.

SUMMARY

Lesson Summary

Summary

You can now solve one-step equations by using inverse operations on both sides. Remember the balance scale like a seesaw!

Practice keeps you sharp for SSAT. You're ready—great work!

Varsity Tutors • SSAT Middle Level • Solve a one-step equation with a variable.