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Learn to find the value of an unknown variable by undoing operations step by step.
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.
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.
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.
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.
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.
A one-step equation needs just one operation to solve. Here are the four types you will see on the ISEE.
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.
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.
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 Mistake | Why It's Wrong | Fix It! |
|---|---|---|
| Operating on only one side | If 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 equations | Dividing 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 inverse | If the equation says x + 3, adding 3 again makes it bigger, not simpler. | Always use the OPPOSITE operation. |
| Forgetting to check | A small arithmetic mistake can give a wrong answer that "looks" right. | Plug your answer back in. If both sides match, you're correct. |
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 Now | What 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 solutions | Equations with fraction and decimal solutions |
| Solving for x | Inequalities: x + 3 > 10 (using the same inverse operation skills!) |
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.