Multi-Step Equations — Solve a multi-step equation with variables on one side. - SHSAT Math

SECTION 1

Where Did Equations Come From?

People have been solving equations for thousands of years. Long before anyone wrote "x," ancient civilizations used clever tricks to figure out unknown amounts. Farmers needed to split harvests fairly. Builders had to calculate how many bricks to order. Traders wanted to know how many coins they were owed. All of these problems boil down to one idea: finding an unknown number when you already know some facts about it.

~1800 BCE • Babylonian Clay Tablets

Ancient Babylonians wrote math problems on clay tablets. They solved equations using words and step-by-step instructions — no letters or symbols!

~250 CE • Diophantus of Alexandria

A Greek mathematician nicknamed the "Father of Algebra" used abbreviations for unknowns. He wrote one of the first algebra textbooks.

~820 CE • Al-Khwarizmi's Algebra

The Persian scholar al-Khwarizmi wrote a book whose title gave us the word "algebra." He described balancing both sides of an equation to solve for unknowns.

1637 • Descartes Uses x, y, z

French mathematician René Descartes popularized using the letters x, y, and z for unknowns. This is the notation we still use today on the SHSAT!

So here is the big question that algebra answers: If you know the result of several operations done to a number, how do you work backward to find that number? That is exactly what multi-step equations are all about.

SECTION 2

Core Principles You Need to Know

Before we dive into solving, let's lock in four key ideas. These rules are the building blocks for every equation you will ever solve.

Balance Rule

An equation is like a balanced scale. Whatever you do to one side, you must do to the other side to keep it equal.

Inverse Operations

Inverse operations (opposite operations) undo each other. Addition undoes subtraction. Multiplication undoes division.

Order of Unwinding

To isolate the variable, reverse the order of operations. Undo addition or subtraction first, then undo multiplication or division.

Simplify First

Before solving, combine like terms and distribute (spread a number across parentheses). A cleaner equation is easier to solve.

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KEY TAKEAWAY

Think of solving an equation like getting dressed in reverse. When you get dressed, you put on socks then shoes. To get undressed, you remove shoes first, then socks. In the same way, you undo the last operation first and work your way backward to the variable.

SECTION 3

Seeing the Steps: A Visual Guide

The diagram below shows how to solve the equation 3x + 5 = 20 using a balance scale model. Each step keeps the scale balanced while peeling away one layer at a time.

Each colored step represents one inverse operation. In Step 1 we undo the "+5" by subtracting 5 from both sides. In Step 2 we undo the "×3" by dividing both sides by 3.

Notice that we always undo addition or subtraction first, then undo multiplication or division. This is because addition and subtraction are the outermost operations — the last things that happen to the variable when you read left to right.

SECTION 4

The Mathematical Framework

A multi-step equation with variables on one side has a general form. Understanding this form helps you recognize what to do no matter how the problem looks.

GENERAL FORM

ax + b = c

a = the number multiplied by the variable (the coefficient), b = the number added or subtracted, c = the number on the other side, and x = the variable you want to find.

STEP 1 — UNDO ADDITION OR SUBTRACTION

ax = c − b

Subtract b from both sides. (If the original has "−b" instead of "+b," you would add b instead.)

STEP 2 — UNDO MULTIPLICATION

x = (c − b) ÷ a

Divide both sides by a. Now x stands alone, and you have your answer!

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What About Parentheses?

Some multi-step equations include parentheses, like 2(x + 3) = 14. In that case, use the distributive property first: multiply the outside number by each term inside the parentheses. Then solve as usual.

DISTRIBUTIVE PROPERTY

a(x + b) = ax + ab

Multiply a by x and multiply a by b. Then rewrite the equation without parentheses.

SECTION 5

Types of Multi-Step Equations You'll See

Not every multi-step equation looks the same. On the SHSAT, you might see several varieties. The diagram below groups the most common types and shows a quick example of each.

The five cards show increasingly complex multi-step equations. Every type uses the same core strategy: simplify, undo addition/subtraction, then undo multiplication/division.

No matter which type you see, the game plan stays the same. Simplify each side first (distribute, combine like terms). Then peel away layers using inverse operations until the variable is by itself.

SECTION 6

Worked Example: Step by Step

Let's work through a full SHSAT-style problem together. Read each step carefully, and notice how we always explain why before we show what.

Solve: 2(3x − 4) + 6 = 26

1
Step 1 — Distribute the 2

Multiply 2 by each term inside the parentheses: 2 × 3x = 6x and 2 × (−4) = −8. Rewrite the equation.

6x − 8 + 6 = 26
2
Step 2 — Combine like terms on the left

The left side has two plain numbers: −8 and +6. Combine them: −8 + 6 = −2.

6x − 2 = 26
3
Step 3 — Undo subtraction (add 2 to both sides)

The variable term 6x has 2 subtracted from it. The inverse of subtraction is addition. Add 2 to both sides: 26 + 2 = 28.

6x = 28
4
Step 4 — Undo multiplication (divide both sides by 6)

The variable x is multiplied by 6. The inverse of multiplication is division. Divide both sides by 6: 28 ÷ 6 = 14/3.

x = 14/3 (or 4⅔)
5
Step 5 — Check your answer

Plug x = 14/3 back in: 2(3 × 14/3 − 4) + 6 = 2(14 − 4) + 6 = 2(10) + 6 = 20 + 6 = 26. ✓ It checks out!

✓ Verified

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SHSAT Tip

On multiple-choice problems, you can sometimes plug each answer choice into the original equation to see which one makes both sides equal. This is called back-substitution and it can save time when the answer choices are simple numbers.

SECTION 7

Common Mistakes and How to Avoid Them

Even strong math students trip up on multi-step equations. The table below lists the most common errors and the fix for each one.

Mistake	What Goes Wrong	How to Fix It
Forgetting to distribute	Writing 2(x + 3) as 2x + 3 instead of 2x + 6.	Multiply the outside number by every term inside the parentheses.
Operating on only one side	Adding 5 to the left but forgetting to add 5 to the right.	Always write the same operation on both sides. Keep the scale balanced!
Wrong sign when distributing a negative	Writing −3(x − 2) as −3x − 6 instead of −3x + 6.	A negative times a negative is a positive. Slow down on the signs.
Dividing before subtracting	Dividing 3x + 6 = 18 by 3 and getting x + 6 = 6 instead of x + 2 = 6.	Undo addition/subtraction first, or if you divide, divide every single term.
Skipping the check	Getting an answer but not verifying it. Small arithmetic errors hide.	Plug your answer back into the original equation and confirm both sides are equal.

Common errors on multi-step equations

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KEY TAKEAWAY

Think of inverse operations like an "undo" button on your phone. If you accidentally delete a photo (subtracted it), you tap undo (add it back). In algebra, every operation has an undo button. Use the undo button in reverse order — last action gets undone first — and you'll always reach the answer.

SECTION 8

From One Side to Both Sides: A Preview

So far, every equation we've looked at has the variable on just one side. But what happens when both sides have a variable? For example: 5x + 3 = 2x + 12. The table below compares the two types.

Feature	Variables on One Side	Variables on Both Sides
Example	4x − 7 = 21	5x + 3 = 2x + 12
First move	Undo addition/subtraction on the variable side	Move variables to one side by subtracting the smaller variable term
Extra step needed?	No — just simplify and solve	Yes — you must gather variable terms first
Core strategy	Inverse operations in reverse order	Same, but with an extra "collect variables" step at the start

Comparison of equation types

The good news? Once you master equations with variables on one side, the jump to both sides is just one extra step. You'll move the smaller variable term to the other side, and then you're right back to the process you already know. Master today's skill and you'll be ready!

SECTION 9

Practice Problems

Try these five problems on your own. They get harder as you go. For each one, write out every step — don't skip ahead!

Question 1: CONCEPTUAL

In the equation 7x + 3 = 31, what is the first step you should take to solve for x? Explain why.

Show answer

The first step is to subtract 3 from both sides, giving 7x = 28. We do this because subtraction is the inverse of the addition that is being done to the 7x term. We always undo addition or subtraction before tackling multiplication. After this step, dividing both sides by 7 gives x = 4.

Question 2: BASIC CALCULATION

Solve for n: 4n − 9 = 15

Show answer

Add 9 to both sides: 4n = 15 + 9 = 24. Then divide both sides by 4: n = 24 ÷ 4 = 6. Check: 4(6) − 9 = 24 − 9 = 15. ✓ The answer is n = 6.

Question 3: INTERMEDIATE

Solve for y: 3(y + 5) − 2 = 25

Show answer

First, distribute the 3: 3y + 15 − 2 = 25. Combine like terms on the left: 3y + 13 = 25. Subtract 13 from both sides: 3y = 12. Divide both sides by 3: y = 4. Check: 3(4 + 5) − 2 = 3(9) − 2 = 27 − 2 = 25. ✓ The answer is y = 4.

Question 4: APPLIED

A movie ticket costs $d. Maria buys 3 tickets and a $7 bag of popcorn. She spends $34 in total. Write and solve an equation to find the price of one ticket.

Show answer

The equation is 3d + 7 = 34. Subtract 7 from both sides: 3d = 27. Divide both sides by 3: d = 9. Each ticket costs $9. Check: 3(9) + 7 = 27 + 7 = 34. ✓ Word problems like this on the SHSAT require you to translate the story into an equation first, then solve using inverse operations.

Question 5: CRITICAL THINKING

Solve for x: 5(2x − 3) + 4x + 1 = 30. Then explain what made this problem harder than the others.

Show answer

Distribute the 5: 10x − 15 + 4x + 1 = 30. Combine like terms: (10x + 4x) + (−15 + 1) = 30, which gives 14x − 14 = 30. Add 14 to both sides: 14x = 44. Divide both sides by 14: x = 44/14, which simplifies to x = 22/7 (about 3.14). Check: 5(2 × 22/7 − 3) + 4(22/7) + 1 = 5(44/7 − 21/7) + 88/7 + 1 = 5(23/7) + 88/7 + 1 = 115/7 + 88/7 + 7/7 = 210/7 = 30. ✓ This problem is harder because it combines distributing, combining like terms with multiple variable terms, and produces a fraction answer. It tests every skill from the lesson at once.

SUMMARY

Lesson Summary

Summary

A multi-step equation with variables on one side follows the general form ax + b = c. To solve it, first simplify each side by using the distributive property and combining like terms. Then use inverse operations in reverse order: undo addition or subtraction first, then undo multiplication or division. This isolates the variable and gives you the solution.

Always remember the Balance Rule — whatever you do to one side, do to the other. Watch out for common traps like forgetting to distribute or incorrect signs with negatives. Finally, always check your answer by plugging it back into the original equation. Master these steps and you'll crush multi-step equation problems on the SHSAT!