Solve a one-step equation for an unknown. - ISEE Lower Level Quantitative Reasoning

SECTION 1

Where Did Equations Come From?

People have been solving puzzles with missing numbers for thousands of years! Long before anyone used letters like x or n, people wrote out word puzzles to find unknown amounts.

An equation (ee-KWAY-zhun) is just a math sentence that says two things are equal. Think of it like a balanced seesaw!

1650 BC • Ancient Egypt

Egyptians wrote puzzles on scrolls asking, "What number plus 5 gives 12?" They solved these with clever guessing.

800 AD • Al-Khwarizmi's Big Idea

A mathematician named Al-Khwarizmi wrote a book about solving equations. He is called the "father of algebra!"

1600s • Letters Replace Words

Mathematicians started using letters like x and n to stand for unknown numbers. This made equations much shorter to write!

Today • Equations Everywhere

We use equations every day — to split a pizza, figure out scores, or plan a trip. The ISEE tests how well you can solve them!

So here's the big question: if someone tells you that a mystery number plus 3 equals 10, how do you figure out the mystery number? That's exactly what this lesson will teach you!

SECTION 2

Core Ideas About Equations

Before we start solving, let's learn the key ideas. These are like the rules of the game!

What Is an Equation?

An equation is a math sentence with an equals sign (=). It says "this side" equals "that side." Example: n + 3 = 7.

What Is an Unknown?

The unknown is the mystery number. We use a letter like x, n, or y to hold its place until we figure it out.

One-Step Means One Move

A one-step equation needs just one math move to solve. You might add, subtract, multiply, or divide — but only once!

Opposite Operations

To undo addition, you subtract. To undo subtraction, you add. To undo multiplication, you divide. To undo division, you multiply. These are called inverse operations.

Keep It Balanced!

Whatever you do to one side of the equation, you must do to the other side too. Think of a seesaw — both sides stay level!

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KEY TAKEAWAY

Think of an equation like a balanced seesaw at the playground. The equals sign is the middle point. If you take something off one side, you must take the same thing off the other side. Otherwise, the seesaw tips over! To find the unknown, use the opposite operation to get the letter all by itself.

SECTION 3

See It: The Balance Model

Let's look at a picture that shows how equations work. Imagine a balance scale, like the ones you might see in a science class. Each side must weigh the same!

The top scale shows the equation n + 3 = 7. We subtract 3 from both sides (shown by the pink arrows). The bottom scale shows the result: n = 4. The scale stays balanced!

See how both sides stay even? That's the most important rule. When we subtracted 3 from the left side, we also subtracted 3 from the right side. The letter n ended up all alone, and we found our answer: n = 4!

SECTION 4

The Math: Inverse Operations

To solve a one-step equation, you use the inverse operation (in-VERS op-er-AY-shun). That's just a fancy way of saying "the opposite." Here are the four types you'll see on the ISEE.

ADDITION EQUATION

n + a = b → n = b − a

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

SUBTRACTION EQUATION

n − a = b → n = b + a

If a number is subtracted from the unknown, add it to both sides to undo it.

MULTIPLICATION EQUATION

a × n = b → n = b ÷ a

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

DIVISION EQUATION

n ÷ a = b → n = b × a

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

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ISEE TIP

On the ISEE, you can check your answer by plugging it back into the equation. If both sides are equal, you got it right! This takes just a few seconds and can save you from silly mistakes.

SECTION 5

Know Your Opposite Operations

The secret to solving one-step equations is knowing which operation undoes which. Let's see all four pairs side by side, then look at a picture that puts it all together.

Each pair of operations undoes the other (shown by the double arrows). The box at the bottom shows the simple 3-step strategy you can use on every one-step equation on the ISEE!

You See This	Do This	Example
n + a number	Subtract that number	n + 6 = 10 → n = 10 − 6 = 4
n − a number	Add that number	n − 4 = 9 → n = 9 + 4 = 13
a number × n	Divide by that number	5 × n = 20 → n = 20 ÷ 5 = 4
n ÷ a number	Multiply by that number	n ÷ 3 = 6 → n = 6 × 3 = 18

Quick reference: which opposite operation to use

SECTION 6

Worked Example: Step by Step

Let's solve a problem together, just like you'd see on the ISEE. Take it one step at a time!

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SAMPLE PROBLEM

Maria had some stickers. She gave 8 stickers to her friend and now has 15 stickers left. How many stickers did Maria start with?

Solving: n − 8 = 15

1
Step 1 — Write the Equation

Maria started with an unknown number of stickers (call it n). She gave away 8, so we subtract 8. She ended up with 15.

n − 8 = 15
2
Step 2 — Find the Operation

The equation shows subtraction (minus 8). The opposite of subtraction is addition.
3
Step 3 — Do the Opposite to Both Sides

Add 8 to both sides of the equation. On the left: n − 8 + 8 = n. On the right: 15 + 8 = 23.

n = 23
4
Step 4 — Check Your Answer

Plug 23 back in: 23 − 8 = 15. ✓ That matches! Maria started with 23 stickers.

Answer: 23 stickers ✓

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REMEMBER

Always check your answer! Plug your number back into the original equation. If both sides match, you nailed it. This only takes a few seconds and is like a free point saver on the ISEE.

SECTION 7

ISEE Strategies & Common Traps

The ISEE loves to test whether you pick the right operation. Here are some strategies and traps to watch out for!

Strategy	Why It Helps
Use the opposite operation	This is the #1 tool. Addition undoes subtraction. Multiplication undoes division. It works every time!
Check by plugging back in	Put your answer in place of the letter. If both sides are equal, you're correct.
Try the answer choices	If you're stuck, try each answer choice in the equation. One of them will make both sides equal!
Never leave a blank	There's no penalty for wrong answers on the ISEE. Always guess if you're not sure!
Watch for tricky wording	"How many more" means subtraction. "Times as many" means multiplication. Read carefully!

Top strategies for one-step equations on the ISEE

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WATCH OUT!

The most common mistake is using the same operation instead of the opposite. For example, if the equation says n + 5 = 12, some students add 5 again instead of subtracting. Remember: to get rid of something, do the opposite!

SECTION 8

From One-Step to Two-Step Equations

Once you master one-step equations, you'll be ready for bigger challenges! Here's a peek at how one-step equations compare to the next level.

Feature	One-Step Equation	Two-Step Equation
How many moves?	Just 1	2 moves
Example	n + 5 = 12	2 × n + 5 = 13
Difficulty	Great starting point!	A bit harder, but same idea
On the ISEE Lower Level?	Yes — very common!	Sometimes, as harder questions

Comparing one-step and two-step equations

The great news is that two-step equations use the exact same skills you're learning now. You just do the opposite operation twice instead of once. Master one-step equations first, and you'll be ready!

SECTION 9

Practice Problems

Time to test your skills! Try each problem on your own first. Remember: find the operation, use the opposite, and check your answer. You've got this!

Question 1: CONCEPTUAL

If n + 4 = 9, what is the value of n?

A. 3
B. 5
C. 9
D. 13

Show answer (B)

The correct answer is B (5). The equation says n + 4 = 9. Since 4 is added to n, we use the opposite operation: subtract 4 from both sides. 9 − 4 = 5, so n = 5. Check: 5 + 4 = 9 ✓. Choice A (3) is wrong because 3 + 4 = 7, not 9. Choice C (9) is wrong — that's just the number on the right side, not the value of n. Choice D (13) is wrong because it adds 4 + 9 instead of subtracting.

Question 2: BASIC CALCULATION

What number makes this equation true? x − 7 = 15

A. 8
B. 12
C. 22
D. 105

Show answer (C)

The correct answer is C (22). The equation has subtraction (−7), so we use the opposite: add 7 to both sides. 15 + 7 = 22, so x = 22. Check: 22 − 7 = 15 ✓. Choice A (8) is wrong because 8 − 7 = 1, not 15. Choice B (12) is wrong because 12 − 7 = 5, not 15. Choice D (105) is wrong — this comes from multiplying 15 × 7 instead of adding.

Question 3: INTERMEDIATE

If 6 × n = 42, what is the value of n?

A. 6
B. 7
C. 8
D. 36

Show answer (B)

The correct answer is B (7). The equation uses multiplication (6 ×), so we use the opposite: divide both sides by 6. 42 ÷ 6 = 7, so n = 7. Check: 6 × 7 = 42 ✓. Choice A (6) is wrong because 6 × 6 = 36, not 42. Choice C (8) is wrong because 6 × 8 = 48, not 42. Choice D (36) is wrong because that comes from subtracting 42 − 6 instead of dividing.

Question 4: APPLIED

Jake has some baseball cards. After his friend gives him 12 more cards, Jake has 35 cards in all. Which equation could you use to find how many cards Jake started with, and what is the answer?

A. 12
B. 23
C. 35
D. 47

Show answer (B)

The correct answer is B (23). Jake started with an unknown number (n) and received 12 more, ending up with 35. The equation is n + 12 = 35. Use the opposite of addition: subtract 12 from both sides. 35 − 12 = 23, so n = 23. Check: 23 + 12 = 35 ✓. Choice A (12) is wrong — that's just the number of cards his friend gave him. Choice C (35) is wrong — that's the total, not the starting amount. Choice D (47) is wrong because it adds 35 + 12, which would give more cards than the total.

Question 5: CRITICAL THINKING

A group of students share a box of markers equally. Each student gets 8 markers, and there are 4 students. Which equation shows the total number of markers (n) in the box, and what is n?

A. 2
B. 12
C. 32
D. 48

Show answer (C)

The correct answer is C (32). The total number of markers (n) was divided equally among 4 students, so n ÷ 4 = 8. Use the opposite of division: multiply both sides by 4. 8 × 4 = 32, so n = 32. Check: 32 ÷ 4 = 8 ✓. Each student gets 8 markers, which matches! Choice A (2) is wrong because 2 ÷ 4 does not equal 8. Choice B (12) is wrong because 12 ÷ 4 = 3, not 8. Choice D (48) is wrong because 48 ÷ 4 = 12, not 8.

SUMMARY

Let's Review!

Summary

An equation is a math sentence with an equals sign. A one-step equation needs just one move to solve. To find the unknown (the mystery number), use the inverse (opposite) operation on both sides. Addition undoes subtraction. Multiplication undoes division.

Follow the 3-step strategy: (1) spot the operation, (2) do the opposite to both sides, and (3) check your answer by plugging it back in. On the ISEE, if you're stuck, try each answer choice in the equation — one will work! And remember, there's no penalty for guessing, so always answer every question. You've got this!