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ISEE Middle Level: Quantitative Reasoning • Algebraic Thinking

Solve one-step or two-step equations.

Learn to find the value of an unknown variable by undoing operations step by step.

Explore the lesson 9 min read
ISEE Middle Level emblem with a school, calculator, and reading passage
Inside this lesson 9 sections 5 questions 9 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 how to find unknown numbers. They used words and sentences instead of symbols. The story of equations is really the story of people finding clever shortcuts to solve everyday problems.

1800 BCE • Babylonian Clay Tablets

Ancient Babylonians wrote math problems on clay tablets. They solved equations using step-by-step instructions, much like recipes.

250 CE • Diophantus of Alexandria

The Greek mathematician Diophantus wrote a famous book called Arithmetica. He used abbreviations for unknowns, an early step toward modern algebra.

820 CE • Al-Khwarizmi's Algebra

The Persian mathematician al-Khwarizmi wrote a book that gave us the word "algebra." He taught methods for balancing equations that we still use today.

1637 • Descartes Uses x, y, z

French mathematician René Descartes popularized using letters like x, y, and z for unknowns. This is the notation you see on the ISEE today!

Today, solving equations is one of the most important skills in math. On the ISEE, you will see one-step and two-step equations in both standard word problems and quantitative comparison questions. The good news? The core idea is simple: undo what has been done to the variable.

SECTION 2

Core Principles of Solving Equations

An equation is a math sentence that says two things are equal. Think of it like a perfectly balanced seesaw. Whatever you do to one side, you must do to the other side to keep it balanced. Here are the key ideas you need to know.

1

The Balance Rule

An equation is like a balanced scale. To keep it balanced, you must perform the same operation on both sides. Add 3 to one side? Add 3 to the other side too.
2

Inverse Operations

Inverse operations are opposites that undo each other. Addition undoes subtraction. Multiplication undoes division. Use them to isolate the variable.
3

Isolate the Variable

Your goal is to get the variable (the letter) all by itself on one side of the equals sign. Then the other side tells you its value.
4

Check Your Answer

Plug your answer back into the original equation. If both sides are equal, you got it right! This is a great habit, especially on timed tests like the ISEE.

SECTION 3

The Balance Model — See It in Action

The diagram below shows how solving a one-step equation works using a balance scale. We start with the equation x + 3 = 7. To isolate x, we subtract 3 from both sides. Watch how the scale stays balanced at every step.

Solving x + 3 = 7 with a Balance ScaleSTEP 1: Startx37x + 3 = 7STEP 2: Subtract 3 from both sidesx4x = 4−3 each sideCHECK: Plug x = 4 back in4 + 3 = 77 = 7 ✓Both sides match, so x = 4 is correct!
The balance scale shows the equation x + 3 = 7. When we subtract 3 from both sides, the scale stays balanced and we discover that x = 4. Always check by substituting your answer back in.

Notice how the key move is using an inverse operation. The equation adds 3 to x, so we subtract 3 to undo it. This same idea works for every one-step equation. If the equation multiplies by 5, you divide by 5. If it subtracts 10, you add 10.

SECTION 4

The Math Behind Solving Equations

One-Step Equations

A one-step equation needs just one operation to solve. Here are the four types you will see on the ISEE.

ADDITION EQUATION

x+a=b→x=b−ax + a = b → x = b − a

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

SUBTRACTION EQUATION

x−a=b→x=b+ax − a = b → x = b + a

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

MULTIPLICATION EQUATION

a×x=b→x=b÷aa × 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×ax ÷ a = b → x = b × a

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

Two-Step Equations

A two-step equation needs two operations. The trick is to go in the right order. Always undo addition or subtraction first, and then undo multiplication or division second.

TWO-STEP EQUATION PATTERN

a×x+b=c→Step1:a×x=c−b→Step2:x=(c−b)÷aa × x + b = c → Step 1: a × x = c − b → Step 2: x = (c − b) ÷ a

First subtract b from both sides to remove the constant. Then divide both sides by a to isolate x.

SECTION 5

Inverse Operations — Your Secret Weapon

The chart below shows the four inverse operation pairs. Every time you see an operation in an equation, use its inverse to undo it. Think of inverse operations as an "undo button" — just like pressing Ctrl+Z on a computer takes back your last action.

Inverse Operations — The Undo PairsPAIR 1Addition⇅Subtractionx + 5 = 12 → subtract 5PAIR 2Multiplication⇅Division3x = 21 → divide by 3TWO-STEP STRATEGYFor 2x + 5 = 13:Step 1: Undo +5Step 2: Undo ×22x = 8 → x = 4 ✓
The two inverse operation pairs are shown at the top. Addition undoes subtraction, and multiplication undoes division. For two-step equations (bottom), always undo addition or subtraction first, then undo multiplication or division.
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ISEE Test Tip

SECTION 6

Step-by-Step Worked Examples

Example 1: One-Step Equation

Worked example

  1. 1

    Step 1 — Identify the Operation

    The equation subtracts 8 from n. The inverse of subtraction is addition.
  2. 2

    Step 2 — Add 8 to Both Sides

    n − 8 + 8 = 15 + 8. The −8 and +8 cancel on the left side.
    n = 23
  3. 3

    Step 3 — Check

    Substitute n = 23 back in: 23 − 8 = 15. ✓ Both sides match!

Example 2: Two-Step Equation

Worked example

  1. 1

    Step 1 — Identify the Operations

    Two things are happening to x: it is multiplied by 3, and then 7 is added. We need to undo both, starting with the addition.
  2. 2

    Step 2 — Subtract 7 from Both Sides

    3x + 7 − 7 = 22 − 7. This removes the +7 from the left side.
    3x = 15
  3. 3

    Step 3 — Divide Both Sides by 3

    3x ÷ 3 = 15 ÷ 3. This isolates x.
    x = 5
  4. 4

    Step 4 — Check

    Substitute x = 5 back in: 3(5) + 7 = 15 + 7 = 22. ✓ It works!

SECTION 7

Common Mistakes & How to Avoid Them

Even strong math students make mistakes with equations. The table below shows the most common errors and how to fix them. Knowing these traps ahead of time will help you avoid them on test day.

Common MistakeWhy It's WrongFix It!
Operating on only one sideIf you subtract 5 from the left but not the right, the equation is no longer balanced.Always do the same thing to BOTH sides.
Wrong order in two-step equationsDividing first in 2x + 4 = 10 gives the wrong answer because 4 is not being divided properly.Undo addition/subtraction first, then multiplication/division.
Using the same operation instead of the inverseIf the equation says x + 3, adding 3 again makes it bigger, not simpler.Always use the OPPOSITE operation.
Forgetting to checkA small arithmetic mistake can give a wrong answer that "looks" right.Plug your answer back in. If both sides match, you're correct.

Common equation-solving mistakes and corrections

SECTION 8

From Simple Equations to Bigger Ideas

One-step and two-step equations are the foundation for all of algebra. Once you master these, you will be ready for more advanced equation types. The table below gives you a preview of what comes next.

What You Know NowWhat Comes Next
One-step equations (x + 5 = 12)Multi-step equations with variables on both sides (2x + 3 = x + 10)
Two-step equations (3x − 4 = 11)Equations with parentheses and the distributive property: 2(x + 3) = 14
Whole number solutionsEquations with fraction and decimal solutions
Solving for xInequalities: x + 3 > 10 (using the same inverse operation skills!)

How today's skills connect to future algebra topics

The great news is that the same inverse operation strategy you learned today works for all of these harder problems. Master the basics now, and you will have a huge head start. For the ISEE Middle Level, one-step and two-step equations are the main focus, so you are learning exactly what you need.

SECTION 9

Practice Problems

ℹ
ISEE Reminder

Question 1: Conceptual

Solve for x: x + 14 = 39
  • A. 15
  • B. 25
  • C. 43
  • D. 53
Show answer (B)
Subtract 14 from both sides: x = 39 - 14 = 25. The correct answer is B. (A) 15 results from computing 39 - 24 instead of 39 - 14, a subtraction error of 10. (C) 43 results from mistakenly computing 39 + 4 — adding the ones digit of 14 rather than subtracting. (D) 53 results from adding instead of subtracting: 39 + 14 = 53.

Question 2: Procedural

Solve for n: 7n = 84
  • A. 11
  • B. 12
  • C. 13
  • D. 77
Show answer (B)
Divide both sides by 7: n=84÷7=12n = 84 \div 7 = 12. The correct answer is B. (A) 11 results from a division error: computing 77÷7=1177 \div 7 = 11 (using 77 instead of 84). (C) 13 results from a division error: computing 91÷7=1391 \div 7 = 13 (confusing 84 with 91). (D) 77 results from subtracting instead of dividing: 84 - 7 = 77.

Question 3: Quantitative Comparison

Given: 3m + 5 = 26 Column A: m Column B: 8
  • A. Column A is greater
  • B. Column B is greater
  • C. The two quantities are equal
  • D. Cannot be determined
Show answer (B)
Solve the equation: subtract 5 from both sides to get 3m = 21, then divide both sides by 3 to get m = 7. Column A = 7 and Column B = 8, so Column B is greater. The correct answer is B. (A) Column A is greater would require m > 8. A student might get m=313≈10.3m = \frac{31}{3} \approx 10.3 by incorrectly adding 5 to 26 before dividing, yielding a value greater than 8. (C) A student who divides 26 by 3 first and rounds down to about 8 might incorrectly conclude the quantities are equal. (D) Both columns are fixed numbers once the equation is solved, so the relationship can be determined.

Question 4: Applied

Maria had some stickers. She gave away 18 stickers and then bought 7 more. She now has 34 stickers. How many stickers did Maria start with?
  • A. 23
  • B. 27
  • C. 45
  • D. 59
Show answer (C)
Let s be the starting number of stickers. Write the equation: s - 18 + 7 = 34, which simplifies to s - 11 = 34. Add 11 to both sides: s = 45. The correct answer is C. (A) 23 results from computing 34 - 18 + 7 = 23 — subtracting 18 from 34 instead of adding it back, then adding 7 instead of subtracting it. (B) 27 results from computing 34 - 7 = 27, reversing the buying step but forgetting to undo the giving-away step. (D) 59 results from adding both amounts to 34: 34 + 18 + 7 = 59, rather than adding 18 and subtracting 7.

Question 5: Quantitative Comparison – Challenge

Given: k4−3=9\frac{k}{4} - 3 = 9 Column A: 2k + 1 Column B: 100
  • A. Column A is greater
  • B. Column B is greater
  • C. The two quantities are equal
  • D. Cannot be determined
Show answer (B)
Solve the equation: add 3 to both sides to get k4=12\frac{k}{4} = 12, then multiply both sides by 4 to get k = 48. Column A = 2(48) + 1 = 97. Column B = 100. Since 97 < 100, Column B is greater. The correct answer is B. (A) A student might mistakenly solve k4=9−3=6\frac{k}{4} = 9 - 3 = 6 by subtracting instead of adding, getting k = 24 and 2(24)+1 = 49, which is still less than 100, so even this error does not lead to choosing A. Alternatively, a student who miscomputes k = 52 (adding 3 then multiplying by 4 as 4×13=524 \times 13 = 52) would get 2(52)+1 = 105 > 100 and incorrectly choose A. (C) A student might round or miscalculate 2k+1 as 100, for example by forgetting the +1 and thinking 2(50) = 100 after missolving for k = 50. (D) Both columns are fixed numbers once the equation is solved, so the relationship can be determined.

Varsity Tutors • ISEE Middle Level • Solve one-step or two-step equations.