Question

problem solving with two-step equations

Answer 1

Solving problems with two-step equations typically involves translating a word problem into a mathematical equation and then solving for the variable. Here’s a general approach, along with some examples:

### Steps to Solve Two-Step Equations:

1. **Understand the problem**: Read the problem carefully to identify what is being asked and what information is given.

2. **Translate into an equation**: Define a variable to represent the unknown quantity. Write an equation based on the relationships and operations described in the problem.

3. **Solve the equation**:
- **Step 1**: Isolate the term with the variable by adding or subtracting.
- **Step 2**: Solve for the variable by multiplying or dividing.

4. **Check your solution**: Substitute your solution back into the original equation or context to ensure it makes sense.

### Examples:

**Example 1:**
A number is multiplied by 3, and then 4 is added to the result, making 19. What is the number?

**Step 1: Define the variable.**
Let \( x \) represent the number.

**Step 2: Write the equation.**
The equation based on the problem is:
\[ 3x + 4 = 19 \]

**Step 3: Solve the equation.**
- First, subtract 4 from both sides:
\[ 3x + 4 - 4 = 19 - 4 \]
\[ 3x = 15 \]

- Next, divide by 3:
\[ x = \frac{15}{3} \]
\[ x = 5 \]

**Step 4: Check the solution.**
Substituting back:
\[ 3(5) + 4 = 15 + 4 = 19 \]
This is correct.

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**Example 2:**
A number decreased by 7 equals 12. What is the number?

**Step 1: Define the variable.**
Let \( y \) represent the number.

**Step 2: Write the equation.**
The equation is:
\[ y - 7 = 12 \]

**Step 3: Solve the equation.**
- Add 7 to both sides:
\[ y - 7 + 7 = 12 + 7 \]
\[ y = 19 \]

**Step 4: Check the solution.**
Substituting back:
\[ 19 - 7 = 12 \]
This is correct.

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### Summary:
By identifying the variable, writing the correct equation, and following the steps to solve it, you can effectively tackle a variety of problems involving two-step equations. Practice with different types of problems to improve your skills!