Solve Equations for an Unknown - ISEE Middle Level Mathematics Achievement

SECTION 1

Why Do We Solve Equations?

People have been solving equations for thousands of years. Ancient civilizations needed to figure out unknown amounts — like how much grain to store or how wide to build a wall. The idea of finding an unknown value (a number you don't know yet) is one of the oldest ideas in math.

Over time, mathematicians developed smarter and faster ways to find these mystery numbers. They invented symbols like variables (letters that stand for unknown numbers) so they could write problems more easily. Today, we use these same tools on the ISEE and in everyday math.

1800 BCE • Babylonian Math Tablets

Ancient Babylonians carved equation problems onto clay tablets. They solved for unknowns using step-by-step methods — long before algebra had a name.

820 CE • Al-Khwarizmi's Book of Algebra

The Persian mathematician al-Khwarizmi wrote a famous book about solving equations. The word "algebra" actually comes from the title of his book!

1637 • Descartes Uses x for Unknowns

French mathematician René Descartes made it popular to use the letter x (and other letters) to represent unknown numbers. This is the same notation you use today.

Today • Equations Everywhere

From video game programming to building bridges, solving for an unknown is one of the most used skills in science, technology, and daily life.

So here's the big question: if you know that some expression equals a number, how do you figure out what the unknown variable is? That's exactly what this lesson will teach you.

SECTION 2

Core Principles of Solving Equations

An equation is a math sentence that says two things are equal. It always has an equals sign (=). Your job when solving an equation is to find the value of the variable that makes the equation true. Here are the key ideas you need.

Balance Rule

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

Inverse Operations

To undo an operation, use its opposite. Addition undoes subtraction, and multiplication undoes division. These opposites are called inverse operations.

Isolate the Variable

Your goal is to get the variable all by itself on one side of the equation. When the variable is alone, you've found the answer.

Check Your Work

Plug your answer back into the original equation. If both sides are equal, you solved it correctly. This is a powerful ISEE test-taking habit.

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KEY TAKEAWAY

Think of an equation like a seesaw at a playground. Both sides must weigh the same to stay level. If you add a 5-pound weight to one side, you have to add a 5-pound weight to the other side too, or the seesaw tips over. Solving an equation means carefully removing weights from both sides until only the variable sits on one end.

SECTION 3

Seeing the Balance

The diagram below shows how solving the equation x + 3 = 7 works like a balance scale. On the left side we have x and 3. On the right side we have 7. To isolate x, we subtract 3 from both sides.

In Step 1, the scale is balanced with x + 3 on the left and 7 on the right. In Step 2, we subtract 3 from both sides. The scale stays balanced, and x is now alone. The answer is x = 4.

Notice that every time we do something to one side, we do the exact same thing to the other side. This is the balance rule in action. If you remember nothing else, remember this: both sides must always stay equal.

SECTION 4

The Math Behind Solving Equations

Let's look at the specific rules you'll use. On the ISEE, you'll see one-step and two-step equations. Here are the inverse operation pairs you need to know.

UNDO ADDITION

If x + a = b, then x = b − a

Subtract a from both sides to undo the addition.

UNDO SUBTRACTION

If x − a = b, then x = b + a

Add a to both sides to undo the subtraction.

UNDO MULTIPLICATION

If a × x = b, then x = b ÷ a

Divide both sides by a to undo the multiplication.

UNDO DIVISION

If x ÷ a = b, then x = b × a

Multiply both sides by a to undo the division.

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ISEE Test Tip

For two-step equations like 3x + 5 = 20, always undo the addition or subtraction first, then undo the multiplication or division. Think: reverse the order of operations!

SECTION 5

Types of Equations You'll See on the ISEE

On the ISEE Middle Level, equations come in two main flavors: one-step equations and two-step equations. Let's break down what each type looks like and how to handle it.

One-step equations need just one inverse operation. Two-step equations need two — always undo the addition or subtraction first, then undo the multiplication or division.

Notice the pattern in the two-step column. You always handle the addition or subtraction step first, then the multiplication or division step second. This is like reversing the order of operations — you "undo" things in the opposite order from how they were built.

SECTION 6

Worked Example: Two-Step Equation

Let's solve a two-step equation step by step, just like you would on the ISEE. Follow along carefully!

Solve: 4x − 9 = 23

1
Step 1 — Identify the operations

Look at what's happening to x. First, x is multiplied by 4. Then 9 is subtracted. We need to undo these in reverse order.
2
Step 2 — Undo the subtraction (add 9 to both sides)

4x − 9 + 9 = 23 + 9. The −9 and +9 cancel on the left side.

4x = 32
3
Step 3 — Undo the multiplication (divide both sides by 4)

4x ÷ 4 = 32 ÷ 4. The 4 cancels on the left side.

x = 8
4
Step 4 — Check the answer

Plug x = 8 back into the original equation: 4(8) − 9 = 32 − 9 = 23. ✓ This matches the right side, so x = 8 is correct!

23 = 23 ✓

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ISEE Strategy: Back-Solving

Since the ISEE gives you four answer choices, you can also solve by plugging each choice into the equation to see which one works. Start with the middle values to save time. This is called back-solving and it's a great backup strategy if you get stuck!

SECTION 7

Common Mistakes & How to Avoid Them

Even strong math students can make errors when solving equations. The ISEE test-makers actually design wrong answer choices based on common mistakes. Here are the top traps and how to dodge them.

Mistake	Example	How to Fix It
Operating on only one side	x + 5 = 12 → x = 12 (forgot to subtract 5)	Always do the same operation to BOTH sides.
Using the wrong inverse	x − 3 = 10 → x = 10 − 3 = 7 (should add, not subtract)	Ask: "What operation do I see?" Then use its opposite.
Wrong order in two-step	2x + 6 = 18 → dividing by 2 first instead of subtracting 6 first	Undo add/subtract FIRST, then undo multiply/divide.
Arithmetic errors	32 ÷ 4 = 6 (should be 8)	Always check by plugging your answer back in.

Common equation-solving mistakes on the ISEE

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KEY TAKEAWAY

Think of wrong answer choices on the ISEE like traps in a video game. The test-makers put them there on purpose. Your best defense is checking your answer — plug it back in and make sure both sides match. That one extra step can be the difference between a right and wrong answer.

SECTION 8

From Simple Equations to Bigger Ideas

The equation-solving skills you're learning now are the foundation for everything you'll do in algebra later on. Here's how this topic connects to what comes next.

What You Learn Now	What Comes Later
One-step equations (x + 5 = 12)	Multi-step equations with variables on both sides
Two-step equations (3x − 4 = 11)	Equations with parentheses and distribution
Inverse operations	Solving inequalities using the same techniques
Checking your answer	Verifying solutions to systems of equations

How today's skills build toward future math

The great news is that the balance rule and inverse operations never change. No matter how complicated an equation gets, you always use the same strategy: undo operations to isolate the variable. Master it now, and future math gets a lot easier.

SECTION 9

Practice Problems

Time to practice! These five problems go from easier to harder. Remember: use inverse operations, keep the equation balanced, and check your work by plugging in.

Question 1: CONCEPTUAL

If n + 7 = 20, what is the value of n?

A. 7
B. 13
C. 20
D. 27

Show answer (B)

The correct answer is B (13). To solve n + 7 = 20, subtract 7 from both sides: n = 20 − 7 = 13. Choice A (7) just picks the number already in the equation without solving. Choice C (20) is the number on the right side and ignores the +7. Choice D (27) adds 7 instead of subtracting it. Check: 13 + 7 = 20 ✓.

Question 2: BASIC CALCULATION

If 6y = 54, what is the value of y?

A. 6
B. 9
C. 48
D. 324

Show answer (B)

The correct answer is B (9). To solve 6y = 54, divide both sides by 6: y = 54 ÷ 6 = 9. Choice A (6) is the coefficient, not the answer. Choice C (48) comes from subtracting 6 instead of dividing. Choice D (324) comes from multiplying 6 × 54 instead of dividing. Check: 6 × 9 = 54 ✓.

Question 3: INTERMEDIATE

If 2x + 5 = 19, what is the value of x?

A. 7
B. 9.5
C. 12
D. 14

Show answer (A)

The correct answer is A (7). First, subtract 5 from both sides: 2x = 19 − 5 = 14. Then divide both sides by 2: x = 14 ÷ 2 = 7. Choice B (9.5) comes from dividing 19 by 2 without first subtracting 5. Choice C (12) comes from subtracting 5 and then subtracting 2 instead of dividing. Choice D (14) is the result after only the first step. Check: 2(7) + 5 = 14 + 5 = 19 ✓.

Question 4: APPLIED

A movie ticket costs $8. Maria bought some tickets and also paid a $5 service fee. She spent $37 in total. Which equation can be used to find t, the number of tickets she bought, and what is the value of t?

A. 8t + 5 = 37; t = 4
B. 8t + 5 = 37; t = 5
C. 8t − 5 = 37; t = 4
D. 8 + 5t = 37; t = 6

Show answer (A)

The correct answer is A. Each ticket costs $8, so t tickets cost 8t dollars. Adding the $5 fee gives 8t + 5 = 37. Subtract 5 from both sides: 8t = 32. Divide by 8: t = 4. Choice B has the right equation but the wrong answer (8 × 5 + 5 = 45, not 37). Choice C uses subtraction instead of addition for the fee. Choice D switches which number gets multiplied by t. Check: 8(4) + 5 = 32 + 5 = 37 ✓.

Question 5: CRITICAL THINKING

If 3(x + 2) = 21, what is the value of x?

A. 5
B. 7
C. 9
D. 19

Show answer (A)

The correct answer is A (5). First, divide both sides by 3 to remove the parentheses: x + 2 = 21 ÷ 3 = 7. Then subtract 2 from both sides: x = 7 − 2 = 5. Choice B (7) is the result of 21 ÷ 3 but forgets to subtract 2. Choice C (9) comes from subtracting 21 − 3 = 18 then dividing by 2. Choice D (19) comes from subtracting 2 from 21 without dividing by 3. Check: 3(5 + 2) = 3(7) = 21 ✓.

SUMMARY

Lesson Summary

Summary

Solving an equation means finding the value of the variable that makes both sides equal. The balance rule says whatever you do to one side, you must do to the other. Use inverse operations — addition undoes subtraction, and multiplication undoes division — to isolate the variable.

For one-step equations, perform a single inverse operation. For two-step equations, undo the addition or subtraction first, then undo the multiplication or division. Always check your answer by plugging it back into the original equation. On the ISEE, you can also use back-solving — testing answer choices — as a backup strategy. You've got this!