Algebra often feels like a puzzle where you’re trying to find a “hidden” number, usually represented by $x$. The key to solving these puzzles is understanding how to move numbers from one side of the equation to the other.
In this guide, we will break down the four fundamental rules of transposition and walk through a multi-step example.
When you move a number across the equals sign ($=$), it performs the opposite operation. Think of it like an “inverse” switch.
| Original Operation | Opposite (Transposed) Operation | General Rule |
| Addition (+) | Subtraction ($-$) | $x + a = b \implies x = b – a$ |
| Subtraction (-) | Addition ($+$) | $x – a = b \implies x = b + a$ |
| Multiplication ($\times$) | Division ($\div$) | $x(a) = b \implies x = \frac{b}{a}$ |
| Division ($\div$) | Multiplication ($\times$) | $\frac{x}{a} = b \implies x = b \times a$ |
Addition: If $x + 2 = 6$, we move the $+2$ to the right side where it becomes $-2$.
$x = 6 – 2$
$x = 4$
Subtraction: If $x – 2 = 6$, we move the $-2$ to the right side where it becomes $+2$.
$x = 6 + 2$
$x = 8$
Multiplication: If $x(2) = 6$ (also written as $2x = 6$), we divide the right side by $2$.
$x = \frac{6}{2}$
$x = 3$
Division: If $\frac{x}{2} = 6$, we multiply the right side by $2$.
$x = 6 \times 2$
$x = 12$
Often, you will see equations that combine these rules. In these cases, you usually handle the addition or subtraction first, then the multiplication or division.
Example1: Solve for $x$ in $2x + 3 = 5$
$x = 1$
Quick Tip: To check your answer, plug your result back into the original equation. For the example above, $2(1) + 3$ equals $5$, so we know our answer is correct!
To solve $2x – 3 = 5$, follow these steps:
Step 1: Move the $-3$ to the other side by changing it to $+3$. This gives you $2x = 5 + 3$, which simplifies to $2x = 8$.
Step 2: Since $2$ is multiplied by $x$, move it to the other side as division. This gives you $x = \frac{8}{2}$.
Final Result: $x = 4$.
To solve $\frac{x}{3} – 3 = 5$, follow these steps:
Step 1: Add $3$ to both sides to isolate the fraction. You get $\frac{x}{3} = 5 + 3$, which simplifies to $\frac{x}{3} = 8$.
Step 2: Since $x$ is divided by $3$, move the $3$ to the other side as multiplication. This gives you $x = 8 \times 3$.
Final Result: $x = 24$.
Sometimes you have multiplication, division, and subtraction all in one problem, such as $\frac{2x}{3} – 3 = 5$:
Step 1 (Add): Move the $-3$ to the other side. You get $\frac{2x}{3} = 5 + 3$, which is $\frac{2x}{3} = 8$.
Step 2 (Multiply): Move the denominator ($3$) to the other side as multiplication. This results in $2x = 8 \times 3$, or $2x = 24$.
Step 3 (Divide): Move the $2$ to the other side as division. You get $x = \frac{24}{2}$.
Final Result: $x = 12$.
Problem: Solve $2x + 3 = x + 5$
When solving for $x$, our goal is to get all the $x$ terms on one side (usually the left) and all the plain numbers on the other (usually the right).
Step 1: Move the terms.
We move the $+3$ to the right side. It becomes $-3$.
We move the $x$ to the left side. It becomes $-x$.
Step 2: Simplify.
$2x$ minus $1x$ leaves us with just $x$.
$5$ minus $3$ leaves us with $2$.
Problem: Solve $5x – 3 – 2x = x + 5$
Sometimes an equation looks more crowded, but the logic remains exactly the same.
Step 1: Transpose variables and constants.
Move the $-3$ to the right side (it becomes $+3$).
Move the $x$ from the right to the left (it becomes $-x$).
Step 2: Combine like terms.
On the left: $5x – 2x$ is $3x$. Then $3x – x$ is $2x$.
On the right: $5 + 3$ is $8$.
Step 3: Isolate $x$.
Since $x$ is being multiplied by $2$, we move the $2$ to the other side by dividing.
Final Answer: