Solving Two-Step Equations

Think of it as a "reverse" button:

• The inverse of addition ($+$) is subtraction ($-$).
• The inverse of subtraction ($-$) is addition ($+$).
• The inverse of multiplication ($\cdot$) is division ($\div$).
• The inverse of division ($\div$) is multiplication ($\cdot$).

Exam Tip: When solving equations, always perform the same operation on both sides of the equals sign to keep the equation balanced. If you subtract $5$ from the left, you must subtract $5$ from the right!

Solving Two-Step Equations: Step-by-Step

A two-step equation requires two distinct inverse operations to reach the solution. The most efficient way to solve these is to follow a specific order: first, remove the constant (the number without a variable), and second, remove the coefficient.

Case 1: Addition and Multiplication

Let's look at a standard example: $3x + 5 = 23$

Step 1: Remove the constant. The number $5$ is being added to the term $3x$. To undo this, we use the inverse operation: subtraction. $3x + 5 - 5 = 23 - 5$ $3x = 18$

Step 2: Remove the coefficient. Now we have $3x = 18$. Since $3$ is multiplying $x$, we use the inverse operation: division. $\frac{3x}{3} = \frac{18}{3}$ $x = 6$

Case 2: Subtraction and Fractions

Sometimes, the coefficient is expressed as a fraction. In these cases, the division is already built into the term.

Example: $\frac{x}{2} - 9 = 11$

Step 1: Remove the constant. The $9$ is being subtracted. We undo this by adding $9$ to both sides. $\frac{x}{2} - 9 + 9 = 11 + 9$ $\frac{x}{2} = 20$

Step 2: Remove the fraction. The term $\frac{x}{2}$ means $x$ is divided by $2$. To undo division, we use multiplication. $\frac{x}{2} \cdot 2 = 20 \cdot 2$ $x = 40$

Practice Problems

Now that you have seen the logic, it is time to test your skills. Try to solve these by following the two-step method: remove the constant first, then the coefficient.

Level 1: Guided Practice

Use the hints provided in your study materials to help you through these initial problems.

1. $2n + 3 = 17$
2. $7y - 4 = -32$
3. $\frac{n}{4} + 5 = 12$
4. $\frac{d}{3} - 4 = -7$
5. $-2y + 35 = -9$
6. $\frac{n}{3} + 14 = 3$

Level 2: Mixed Operations

These combine different types of operations. Watch your signs (positive and negative numbers)!

7. $-5k - 2 = -12$
8. $9w - 7 = 11$
9. $\frac{x}{4} + 5 = -8$
10. $\frac{x}{3} - 10 = -6$
11. $\frac{x}{2} + 7 = -2$
12. $\frac{x}{5} - 12 = -10$

Level 3: Advanced (No Hints)

Challenge yourself by solving these without any instructional aids. Focus on the order of operations in reverse.

13. $3g + 15 = -12$
14. $-4f + 8 = 20$
15. $\frac{x}{15} + 13 = 0$
16. $23 - 6d = -7$