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  1. SSAT Middle Level Quantitative

  3. Solve a multi-step equation with a variable.

SSAT Middle Level • Quantitative

Solve a multi-step equation with a variable.

Undo operations step by step to find what the variable equals, like unlocking a puzzle.

SECTION 1

Historical Context & Motivation

Long ago, people needed to solve problems like sharing land or trading goods. Equations (math sentences with a missing number) helped them find answers. Today, solving multi-step equations builds on this to handle tougher puzzles.

2000 BCE
Babylonians used early equations on clay tablets for farming problems.
820 CE
Al-Khwarizmi
A Persian mathematician wrote the first algebra book, meaning 'restoration and completion'.
1500s
European scholars made algebra symbols like = and x for unknowns.
Today
SSAT students use these to solve real-world math challenges.

These steps led to multi-step equations. They let you find one unknown in a busy math sentence.

SECTION 2

Core Principles & Definitions

An equation is like a balanced scale. A variable (like x) is the mystery number you solve for. Multi-step means more than one operation to undo.

1

Balance Rule

Do the same to both sides, like adding weights equally.
2

Inverse Operations

Undo addition with subtraction, multiplication with division.
3

Order Matters

Undo in reverse PEMDAS order: parentheses, exponents, multiply/divide, add/subtract.
4

Isolate Variable

Get x alone on one side. Check by plugging back in.
✦ KEY TAKEAWAY
KEY TAKEAWAY
SECTION 3

Visual Explanation

Balance Scale for 3x + 5 = 203x + 5x box + 5=20known numberBalanced!Undo from outside in to keep balance.
The scale tips if unbalanced. 3x + 5 must equal 20 exactly.

See how the left side has the variable x multiplied by 3 and then added 5? You undo addition first, then division, to keep the scale level.

SECTION 4

Mathematical Framework

Follow these steps every time. Start by undoing the last operation done to x.

GENERAL FORM
ax + b = c
a, b, c are numbers you know. x is unknown.
STEP 1: SUBTRACT b
ax = c − b
STEP 2: DIVIDE BY a
x = (c − b) ÷ a
  • Always do the same to both sides.
  • Check: Plug x back into original equation.
  • Fractions ok, like x = 5/3.
SECTION 5

Detailed Step-by-Step Breakdown

Solving a Multi-Step Equation Follow each transformation to isolate the variable 1 3x + 5 = 20 📝 Original Equation 2 3x + 5 − 5 = 20 − 5 ➖ Subtract 5 from both 3 3x = 15 ✨ Simplify both sides 4 3x ÷ 3 = 15 ÷ 3 ➗ Divide both by 3 ✓ x = 5 🎯 Solution!
Each box shows one step. Arrows point down as you simplify.

Notice the colors match operations. Subtract first to undo add, then divide to undo multiply.

SECTION 6

Worked Example

Let's solve 2(x − 3) = 10. You can do this!

Step 1: Distribute 2

2x − 6 = 10 (multiply each inside term)

Step 2: Add 6 to both sides

2x − 6 + 6 = 10 + 6
2x = 16

Step 3: Divide by 2

2x ÷ 2 = 16 ÷ 2
x = 8

Step 4: Check

2(8 − 3) = 2(5) = 10. Yes!
SECTION 7

Strengths & Common Errors

StrengthCommon MistakeFix
✦ KEY TAKEAWAY
SECTION 8

Connection to Advanced Equations

This LessonAdvanced

Master this, then tackle equations with two x's or more variables. You're on your way!

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL
What do you do first to solve 4x + 7 = 19? A) Divide by 4 B) Subtract 7 C) Add 7 D) Multiply by 4 E) Subtract 4
PROBLEM 2 — BASIC CALCULATION
Solve 5x − 3 = 12. A) x = 3 B) x = 15 C) x = 2 D) x = 5 E) x = 1
PROBLEM 3 — INTERMEDIATE
Solve 2x + 8 = 20. A) x = 6 B) x = 14 C) x = 4 D) x = 12 E) x = 28
PROBLEM 4 — APPLIED
A game costs 3 tokens plus 4 more to play. Total 19 tokens. Equation: 3x + 4 = 19. x = ? A) 4 B) 5 C) 23 D) 15 E) 3
PROBLEM 5 — CRITICAL THINKING
Solve 4(x − 2) + 3 = 11. A) x = 2.5 B) x = 3 C) x = 1 D) x = 10 E) x = 0
SUMMARY

Lesson Summary

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