Find out x, if y=mx+b. Do it step by step 

To solve for \( x \) in the equation \( y = mx + b \), we'll go through a step-by-step process using algebraic manipulation. This equation represents a linear relationship, where \( m \) is the slope and \( b \) is the y-intercept. Let's isolate \( x \) on one side of the equation.

### Equation:

\[
y = mx + b
\]

### Step-by-Step Solution:

#### Step 1: Subtract \( b \) from Both Sides

We want to isolate the term with \( x \) by removing the constant \( b \) from the right side of the equation. Subtract \( b \) from both sides:

\[
y - b = mx
\]

#### Step 2: Divide Both Sides by \( m \)

Now, to isolate \( x \), divide both sides of the equation by the coefficient \( m \):

\[
\frac{y - b}{m} = x
\]

#### Step 3: Write the Solution

Thus, the expression for \( x \) in terms of \( y \), \( m \), and \( b \) is:

\[
x = \frac{y - b}{m}
\]

This formula provides \( x \) in terms of the given variables \( y \), \( m \), and \( b \).

### Example:

Let's use a specific example to see how this formula works:

#### Given:

- \( y = 10 \)
- \( m = 2 \)
- \( b = 3 \)

#### Solution:

Using the formula \( x = \frac{y - b}{m} \):

\[
x = \frac{10 - 3}{2}
\]

Calculate:

\[
x = \frac{7}{2} = 3.5
\]

So, for \( y = 10 \), \( m = 2 \), and \( b = 3 \), the value of \( x \) is 3.5.

### Verification:

To ensure the solution is correct, substitute \( x = 3.5 \) back into the original equation:

\[
y = mx + b = 2 \times 3.5 + 3
\]

Calculate:

\[
y = 7 + 3 = 10
\]

The calculated \( y \) value matches the given \( y \), verifying the solution.

---

### Summary:

The derived formula \( x = \frac{y - b}{m} \) successfully isolates \( x \) in terms of \( y \), \( m \), and \( b \). Always check that \( m \neq 0\) to avoid division by zero.

### Questions and Explorations:

1. **What happens if \( m = 0 \) in the equation?**
2. **Can you solve for \( y \) in terms of \( x \)?**
3. **How would you express \( b \) in terms of \( x \), \( y \), and \( m \)?**
4. **What does the slope \( m \) represent geometrically?**
5. **Can you find \( x \) if \( y \), \( m \), and \( b \) are functions themselves?**

### Tip:

When rearranging equations, carefully follow each algebraic step and check each transformation to ensure accuracy.

Learn how to find x in the equation y = mx + b with a detailed step-by-step solution. This problem covers key algebraic concepts, including linear equations and the slope-intercept form. Suitable for students in Grades 8-10.

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