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HSPT Math • Mathematics

Solve Linear Equations

Learn to find the unknown value that makes an equation true — a must-know skill for the HSPT.

Explore the lesson 9 min read
HSPT emblem with a school, checked assessment, and reasoning brain
Inside this lesson 10 sections 5 questions 8 figures

SECTION 1

Where Did Equations Come From?

People have been solving equations for thousands of years. Long before we used letters like x and y, ancient civilizations figured out ways to find unknown numbers. They needed this skill to build buildings, trade goods, and measure land.

A linear equation (an equation where the variable is not squared or cubed) is the simplest kind. Solving one means finding the number that makes both sides equal. Let's see how this idea grew over time.

1800 BCE • Babylonian Clay Tablets

Ancient Babylonians wrote word problems on clay tablets. They described unknowns as "the thing" and solved them step by step — much like we do today.

250 CE • Diophantus of Alexandria

The Greek mathematician Diophantus used short symbols instead of full words. He is often called the "Father of Algebra."

820 CE • Al-Khwarizmi's Algebra Book

The Persian scholar al-Khwarizmi wrote a famous book that gave us the word "algebra." He showed clear rules for balancing equations.

1637 • Descartes Uses x, y, z

French mathematician René Descartes started using the letters x, y, and z for unknowns. This is the notation (way of writing) we still use in class today.

So the big question people kept asking was: "What number makes this statement true?" That is exactly what you will learn to answer in this lesson.

SECTION 2

Core Principles of Linear Equations

Before you start solving, you need a few key ideas. Think of an equation like a balanced scale. Whatever you do to one side, you must do to the other side to keep it balanced.

1

Equation = Balance

The equals sign (=) means both sides have the same value. Your job is to keep them equal while you work.
2

Inverse Operations

To undo addition, subtract. To undo multiplication, divide. These opposite moves are called inverse operations.
3

Isolate the Variable

Your goal is to get the variable (the letter) alone on one side. Move everything else to the other side.
4

Do the Same to Both Sides

Every operation you perform must happen on both sides of the equation. This is the golden rule.
5

Check Your Answer

Plug your answer back into the original equation. If both sides are equal, you got it right!
✦

KEY TAKEAWAY

Imagine a seesaw with two kids of equal weight — it's perfectly balanced. If you add a backpack to one side, you must add the same weight to the other side, or the seesaw tips. An equation works the same way: always do the same thing to both sides to keep it balanced.

SECTION 3

Visualizing the Balance

The diagram below shows how solving the equation x + 3 = 7 is like removing weight from both sides of a balanced scale until the variable stands alone.

Solving x + 3 = 7 as a Balance ScaleSTEP 1: Start balancedx37Left sideRight sideSTEP 2: Subtract 3 from both sidesx4Left sideRight side−3 removedResult: x = 4Check: 4 + 3 = 7 ✓
In Step 1 the scale is balanced: x + 3 on the left equals 7 on the right. In Step 2 we subtract 3 from both sides, leaving x = 4.

Notice that when we removed the pink block (3) from the left side, we also removed 3 from the right side. The scale stayed level. That is the golden rule in action: do the same thing to both sides.

SECTION 4

The Mathematical Framework

A linear equation has one variable (usually x) that is not raised to a power higher than 1. Here is the general form.

GENERAL FORM

ax+b=cax + b = c

a = the number multiplied by x (the coefficient), b = a number added or subtracted, c = the number on the other side of the equals sign.

To solve, use inverse operations in two stages.

STEP A — UNDO ADDITION / SUBTRACTION

ax+b−b=c−b→ax=c−bax + b − b = c − b → ax = c − b

Subtract b from both sides to remove the constant term.

STEP B — UNDO MULTIPLICATION

ax÷a=(c−b)÷a→x=(c−b)/aax ÷ a = (c − b) ÷ a → x = (c − b) / a

Divide both sides by a to get x alone.

💡

Order Matters!

Always undo addition or subtraction first, then undo multiplication or division. Think of it as "peeling off" layers — the outer layer comes off first.

SECTION 5

Types of Linear Equations You'll See

On the HSPT, linear equations come in several flavors. The diagram below shows four common types, from the simplest one-step problem to equations with variables on both sides.

Types of Linear EquationsTYPE 1 — ONE-STEPx + 5 = 12Undo one operation (subtract 5).TYPE 2 — TWO-STEP3x − 4 = 11Undo subtraction, then division.TYPE 3 — DISTRIBUTIVE PROPERTY2(x + 3) = 14Distribute first, then solve two-step.TYPE 4 — VARIABLE ON BOTH SIDES5x + 2 = 3x + 10Move variables to one side first.STRATEGY SUMMARY1. Distribute if there are parentheses.2. Combine like terms on each side.3. Move variable terms to one side, constants to the other.4. Divide to isolate the variable.
The four main types of linear equations you will encounter on the HSPT. Arrows show how difficulty increases. The green box at the bottom summarizes the general strategy.
TypeExampleSteps Needed
One-stepx − 9 = 4Add 9 to both sides → x = 13
Two-step2x + 6 = 18Subtract 6, then divide by 2 → x = 6
Distributive4(x − 1) = 20Distribute 4, add 4, divide by 4 → x = 6
Variables both sides7x + 1 = 4x + 16Subtract 4x, subtract 1, divide by 3 → x = 5

Common linear equation types with quick solutions

SECTION 6

Worked Example — Step by Step

Let's solve a two-step equation from start to finish. Follow each step carefully.

Solve: 5x − 8 = 22

  1. 1

    Step 1 — Write the equation

    Start with the original equation: 5x − 8 = 22. Our goal is to get x by itself.
  2. 2

    Step 2 — Undo subtraction (add 8 to both sides)

    The 8 is being subtracted from the x-term. To undo that, add 8 to both sides: 5x − 8 + 8 = 22 + 8.
    5x = 30
  3. 3

    Step 3 — Undo multiplication (divide both sides by 5)

    Now x is being multiplied by 5. Divide both sides by 5: 5x ÷ 5 = 30 ÷ 5.
    x = 6
  4. 4

    Step 4 — Check the answer

    Plug x = 6 back into the original equation: 5(6) − 8 = 30 − 8 = 22. The left side equals the right side, so our answer is correct!
    22 = 22 ✓
✦

REMEMBER

Always check your answer. It only takes a few seconds: substitute your number back in and see if both sides match. On the HSPT, this can help you catch careless mistakes and earn more points.

SECTION 7

Common Mistakes & How to Avoid Them

Even strong math students make errors when solving equations. The table below lists the most common mistakes and shows you how to fix them.

MistakeWhy It HappensHow to Fix It
Forgetting to do the same thing to both sidesRushing through the problemWrite each operation on both sides before simplifying
Sign errors (mixing up + and −)Careless subtraction or adding negativesRewrite subtraction as adding a negative: 5 − 8 = 5 + (−8)
Dividing before subtractingWrong order of inverse operationsUndo addition/subtraction first, then multiplication/division
Forgetting to distributeSkipping the parentheses stepMultiply the outside number by every term inside the parentheses
Not checking the answerRunning out of time or feeling confidentBudget 15–20 seconds to plug your answer back in

Top 5 equation-solving mistakes and remedies

🎯

TEST-DAY TIP

On the HSPT, many wrong answer choices are designed to match common mistakes. For example, if the correct answer to 3x + 6 = 21 is x = 5, one wrong choice might be x = 9 (from dividing before subtracting). Knowing the traps helps you avoid them.

SECTION 8

From Linear Equations to Bigger Ideas

Once you master linear equations, you're ready for more advanced topics in high school math. The table below shows how this skill connects to what comes next.

What You Know NowWhat Comes Next
Solve one equation with one variable (e.g., 2x + 3 = 11)Solve a system of two equations with two variables
Variable raised to the 1st powerQuadratic equations where x is raised to the 2nd power
Equations with numbers onlyInequalities that use < or > instead of =
Finding x as a numberGraphing the equation as a straight line on a coordinate plane

How linear equations connect to future math topics

Every one of these advanced topics uses the same balance principle you learned here. If you can solve a linear equation, you have the foundation for all of algebra.

SECTION 9

Practice Problems

Try these five problems on your own. They get harder as you go. After each one, check the answer and read the explanation.

Question 1: CONCEPTUAL

True or false: To solve the equation x + 10 = 25, you should add 10 to both sides. Explain your reasoning.
Show answer
False. Since 10 is being added to x, you need to use the inverse operation and subtract 10 from both sides. That gives x = 25 − 10 = 15. Adding 10 would move you further from the answer, not closer. Always undo the operation that is applied to the variable.

Question 2: BASIC CALCULATION

Solve for x: 4x + 7 = 31
Show answer
First, subtract 7 from both sides: 4x = 31 − 7 = 24. Next, divide both sides by 4: x = 24 ÷ 4 = 6. Check: 4(6) + 7 = 24 + 7 = 31. ✓ This is a classic two-step equation. Remember to undo addition before division.

Question 3: INTERMEDIATE

Solve for x: 3(x − 2) = 21
Show answer
Distribute the 3: 3x − 6 = 21. Add 6 to both sides: 3x = 27. Divide both sides by 3: x = 9. Check: 3(9 − 2) = 3(7) = 21. ✓ The key step is distributing the 3 to both terms inside the parentheses before doing anything else.

Question 4: APPLIED

Maria has $15 in her wallet. She earns $8 for every lawn she mows. After mowing some lawns, she has $63 in total. Write and solve an equation to find how many lawns she mowed.
Show answer
Let x = the number of lawns. The equation is 8x + 15 = 63. Subtract 15 from both sides: 8x = 48. Divide both sides by 8: x = 6. Maria mowed 6 lawns. Check: 8(6) + 15 = 48 + 15 = 63. ✓ In word problems, the first step is always to define your variable and set up the equation before solving.

Question 5: CRITICAL THINKING

Solve for x: 6x + 5 = 2x + 29. Then explain why you need to move the variable terms to the same side.
Show answer
Subtract 2x from both sides: 4x + 5 = 29. Subtract 5 from both sides: 4x = 24. Divide by 4: x = 6. Check: 6(6) + 5 = 41 and 2(6) + 29 = 41. ✓ You need to collect the variable terms on one side because you cannot isolate x if it appears in two places. Moving all the x-terms together lets you combine them into a single term, which you can then solve with simple two-step methods.

SUMMARY

Lesson Summary

Summary

A linear equation is a statement that two expressions are equal, where the variable has an exponent of 1. To solve one, use inverse operations — addition undoes subtraction, and division undoes multiplication. The golden rule is to perform every operation on both sides of the equation so the balance is maintained.

For one-step equations, a single inverse operation finds x. For two-step equations, undo addition or subtraction first, then undo multiplication or division. When parentheses appear, distribute before solving. When the variable appears on both sides, move all variable terms to one side first. Always check your answer by substituting it back into the original equation.

Varsity Tutors • HSPT Math • Solve Linear Equations