One- and Two-Step Equations — Solve a one-step or two-step...

SECTION 1

Where Did Equations Come From?

People have been solving equations for thousands of years. Long before anyone wrote "x" on paper, ancient civilizations needed to figure out unknown amounts. A farmer might ask, "I harvested some wheat. After giving away 10 bags, I have 25 left. How much did I start with?" That question is really an equation (a math statement that says two things are equal) in disguise.

~1800 BCE

Babylonian Clay Tablets

Ancient Babylonians carved word problems onto clay tablets. They solved equations without any symbols — just words and numbers!

~250 CE

Diophantus of Alexandria

A Greek mathematician wrote a book of number puzzles. He used a special symbol for the unknown quantity, one of the first steps toward modern algebra.

~820 CE

Al-Khwarizmi's Breakthrough

A Persian scholar named al-Khwarizmi wrote the first textbook about solving equations. The word "algebra" comes from the Arabic title of his book!

1600s

Modern Notation Takes Shape

Mathematicians in Europe started using letters like x, y, and z for unknowns, plus the equals sign (=). This is the style we still use today.

So when you solve an equation on the SHSAT, you are using the same ideas that people figured out thousands of years ago. The big question is: how do you "undo" the math around a variable to find its value? That's exactly what this lesson will teach you.

SECTION 2

Core Principles of Solving Equations

Before you start solving, you need to understand a few big ideas. These ideas are the foundation for every equation you will ever solve.

An Equation Is a Balance

Think of an equation like a balanced seesaw. The equals sign (=) is the middle. Whatever you do to one side, you must do to the other side to keep it balanced.

Inverse Operations Undo Each Other

Addition and subtraction are opposites. Multiplication and division are opposites. You use inverse operations (opposite operations) to "peel away" the math around the variable.

Isolate the Variable

Your goal is to get the variable (the letter) by itself on one side of the equation. When you do that, the other side tells you the answer.

Check Your Work

After you find the answer, plug it back into the original equation. If both sides are equal, you got it right!

✦ KEY TAKEAWAY

Think of solving an equation like unwrapping a present. The variable is the gift inside, and the numbers and operations are the wrapping paper. You carefully remove one layer at a time — using the opposite action for each layer — until the gift (the answer) is revealed. The key rule? Whatever you do to one side, do the same to the other.

SECTION 3

Visualizing One-Step Equations

The best way to understand equations is to picture a balance scale. Each side of the equation sits on one pan of the scale. The diagram below shows how solving x + 5 = 12 works visually.

The balance scale shows three steps: start with the equation balanced, subtract 5 from both sides (the inverse of adding 5), and arrive at x = 7.

Notice how every step keeps the scale balanced. We subtracted 5 from both sides, not just one. If you only changed one side, the scale would tip — and the equation would no longer be true. This "keep it balanced" rule is the single most important idea when solving any equation.

SECTION 4

The Math Behind Solving Equations

Let's look at the formulas and rules you need. There are four types of one-step equations, one for each operation. Then two-step equations combine two of these operations together.

One-Step Equations

ADDITION EQUATION

x + a = b → x = b − a

If a number is added to x, subtract it from both sides to undo it.

SUBTRACTION EQUATION

x − a = b → x = b + a

If a number is subtracted from x, add it to both sides.

MULTIPLICATION EQUATION

a × x = b → x = b ÷ a

If x is multiplied by a number, divide both sides by that number.

DIVISION EQUATION

x ÷ a = b → x = b × a

If x is divided by a number, multiply both sides by that number.

Two-Step Equations

TWO-STEP GENERAL FORM

a × x + b = c

First, undo the addition or subtraction (deal with b). Then, undo the multiplication or division (deal with a). Think: undo in reverse order of operations.

💡 SHSAT Tip

On the SHSAT, two-step equations are very common. Remember: handle the addition or subtraction first, then handle the multiplication or division second. This "reverse order of operations" trick works every time.

SECTION 5

Types of Equations You'll See

Let's organize the different equation types you might face on the SHSAT. The diagram below shows how one-step and two-step equations are related.

This map shows the main types of equations. One-step equations use a single inverse operation. Two-step equations require you to undo two operations in reverse order.

Common equation types with solving strategies

Equation Type	Example	Steps to Solve
One-step (addition)	x + 8 = 15	Subtract 8 from both sides → x = 7
One-step (subtraction)	x − 4 = 10	Add 4 to both sides → x = 14
One-step (multiplication)	5x = 35	Divide both sides by 5 → x = 7
One-step (division)	x ÷ 3 = 9	Multiply both sides by 3 → x = 27
Two-step	3x + 7 = 22	Subtract 7 → 3x = 15, then divide by 3 → x = 5
Two-step	x/2 − 6 = 4	Add 6 → x/2 = 10, then multiply by 2 → x = 20

SECTION 6

Worked Examples: Step by Step

Example 1: One-Step Equation

Solve: 7x = 42

Step 1 — Identify the OperationThe variable x is being multiplied by 7. To undo multiplication, we use its inverse: division.

Step 2 — Divide Both Sides by 77x ÷ 7 = 42 ÷ 7. On the left side, 7 ÷ 7 cancels out, leaving just x. On the right side, 42 ÷ 7 = 6.

x = 6

Step 3 — Check Your AnswerPlug x = 6 back in: 7 × 6 = 42. That matches the right side. ✓

Example 2: Two-Step Equation

Solve: 4n − 9 = 19

Step 1 — Identify the Two OperationsThe variable n is being multiplied by 4 and then 9 is subtracted. We need to undo these in reverse order: first undo the subtraction, then undo the multiplication.

Step 2 — Add 9 to Both Sides4n − 9 + 9 = 19 + 9. The left side simplifies to 4n. The right side becomes 28.

4n = 28

Step 3 — Divide Both Sides by 44n ÷ 4 = 28 ÷ 4. The left side simplifies to n. The right side becomes 7.

n = 7

Step 4 — Check Your AnswerPlug n = 7 back in: 4 × 7 − 9 = 28 − 9 = 19. That matches! ✓

Example 3: Two-Step with a Fraction

Solve: y/3 + 5 = 11

Step 1 — Undo the Addition FirstSubtract 5 from both sides: y/3 + 5 − 5 = 11 − 5. This gives us y/3 = 6.

y/3 = 6

Step 2 — Undo the DivisionSince y is divided by 3, multiply both sides by 3: (y/3) × 3 = 6 × 3. The left side becomes y. The right side becomes 18.

y = 18

Step 3 — CheckPlug in: 18/3 + 5 = 6 + 5 = 11. ✓

SECTION 7

Common Mistakes and How to Avoid Them

Even strong students make mistakes on equations. Let's look at the most common errors and how to dodge them on test day.

Common equation-solving mistakes

Mistake	Example of the Error	How to Fix It
Only changing one side	x + 3 = 10 → subtract 3 from left only → x = 10 (wrong!)	Always do the same operation to BOTH sides. x = 10 − 3 = 7.
Wrong order in two-step	2x + 4 = 12 → divide by 2 first → wrong path	Handle addition/subtraction FIRST, then multiplication/division.
Sign errors with negatives	x − 5 = 3 → subtract 5 instead of adding → x = −2 (wrong!)	Ask: what undoes "minus 5"? Adding 5! x = 3 + 5 = 8.
Forgetting to check	Getting an answer but not verifying	Plug your answer back into the original equation. It takes 10 seconds and catches mistakes.

🎯 KEY TAKEAWAY

Think of a two-step equation like getting dressed in the morning. You put on socks first, then shoes. To undo it (get undressed), you take off the shoes first, then the socks — reverse order. Equations work the same way. Undo the last thing that was done to the variable first.

SECTION 8

What Comes After One- and Two-Step Equations?

Mastering one- and two-step equations is a stepping stone. Here is how these ideas grow as you move into more advanced math.

How equation skills build over time

What You Know Now	What's Coming Next
One-step: x + 5 = 12	Multi-step with variables on both sides: 3x + 5 = x + 17
Two-step: 2x − 3 = 9	Equations with parentheses: 2(x − 3) = 10 (distributive property)
Whole number solutions	Fraction and decimal solutions
Single equations	Systems of equations (two equations, two unknowns)

The good news? Every future equation still uses the same two rules you learned today: keep the equation balanced, and use inverse operations. If you have those ideas down cold, you are ready for whatever comes next.

📝 SHSAT Preview

On the actual SHSAT, equation problems often appear inside word problems. You will need to translate words into an equation first, then solve. For example, "Three times a number, decreased by 7, is 20" becomes 3x − 7 = 20. Practice turning sentences into equations!

SECTION 9

Practice Problems

Try these five problems. They start easy and get harder. For each one, solve for the variable and check your answer.

PROBLEM 1 — CONCEPTUAL

What is the inverse operation of multiplication? If you see the equation 6x = 48, what operation should you use on both sides to solve for x?

PROBLEM 2 — BASIC CALCULATION

Solve for x: x − 14 = 23

PROBLEM 3 — INTERMEDIATE

Solve for n: 5n + 12 = 47

PROBLEM 4 — APPLIED

Maria has some money saved. She earns $8 per hour babysitting. After working for some hours, she added her earnings to her savings and had $53 total. If she worked 4 hours, how much did she already have saved? Write and solve an equation.

PROBLEM 5 — CRITICAL THINKING

A number is tripled and then 8 is subtracted. The result is the same as adding 13 to the original number. What is the number? (Hint: set up an equation with the variable on both sides, then use what you know.)

SUMMARY

Lesson Summary

An equation is a math sentence with an equals sign. To solve it, you isolate the variable by using inverse operations — addition undoes subtraction, and multiplication undoes division. The golden rule is: whatever you do to one side, do the same to the other.

For one-step equations, use a single inverse operation. For two-step equations, undo operations in reverse order — handle addition or subtraction first, then multiplication or division. Always check your answer by plugging it back into the original equation. These skills are essential for the SHSAT and for every math course to come.