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Learn how to find the mystery number hiding behind a variable using inverse operations.
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.
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.
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.
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.
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.
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.
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.
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.
Let's solve a two-step equation step by step, just like you would on the ISEE. Follow along carefully!
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. |
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 |
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.
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.
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!