How to Solve One-Variable Equations by Keeping the Balance

What this lesson covers

An equation is two expressions with an equals sign between them. Solving it means finding the value of $x$ that makes it true. This lesson teaches how to do that for equations like $3x + 2 = 14$, $\frac{2}{3}x = 12$ and $2(x + 5) - x = 17$.

On the ISEE Middle Level math sections, equations come as multiple-choice questions and as quantitative comparisons, with no calculator. The lesson pictures an equation as a balanced hanger, with bags of marbles for $x$ and loose marbles for the numbers.

The method

The lesson teaches two methods. The first is "keep the balance."

1. Do the same to both sides. On the hanger, taking the same number of marbles off both sides keeps it balanced. So does splitting both sides into the same number of equal groups.
2. Undo in reverse order. Look at what was done to $x$, and undo the last step first.
3. Check in the original equation. Put your answer back into the equation you started with.
4. Simplify a messy side first. If a side has parentheses or like terms, distribute and combine like terms. Then keep the balance.

The second method is "find the whole as an equation." If $p \times x = q$, then $x = q \div p$. When $p$ is a fraction or a percent, this is finding the whole from a part.

When a question asks for an expression instead of $x$, try doing the same to both sides to reach it. If $2x + 3 = 11$, doubling both sides gives $4x + 6 = 22$.

Worked example

A hanger is balanced. On the left side hang 4 bags of marbles and 3 loose marbles. On the right side hang 27 loose marbles. Every bag holds the same number of marbles. How many marbles are in one bag?

Let $x$ be the number of marbles in one bag. The left side holds $4x + 3$ marbles, and the right side holds 27. The hanger is balanced, so $4x + 3 = 27$.

What was done to $x$? It was multiplied by 4, and then 3 was added. Undo in reverse order, so the 3 goes first. Take 3 marbles off both sides: $4x = 24$. Now split both sides into 4 equal groups. Each group is one bag on the left and 6 marbles on the right, so $x = 6$.

Check in the original equation: $4 \times 6 + 3 = 24 + 3 = 27$. Both sides are 27, so one bag holds 6 marbles.

Why take the loose marbles off first? They aren't inside the bags. Splitting first would mean sharing 3 marbles among 4 groups, which gets messy.

The common mistake

The lesson's common mistake is dividing only one term. Take $2x + 6 = 14$. A student divides both sides by 2 and writes $x + 6 = 7$, so $x = 1$.

The check catches it: $2 \times 1 + 6 = 8$, and 8 is not 14.

Dividing both sides by 2 means dividing every term. The $2x$ becomes $x$, the 6 becomes 3, and the 14 becomes 7. So $x + 3 = 7$, and $x = 4$. Undoing in reverse order gives the same answer: take 6 off both sides to get $2x = 8$, then divide by 2. Check: $2 \times 4 + 6 = 14$.

"Both sides" means every term on both sides. And the check caught the mistake, which is why checking is part of the method.