If f is a linear function and `f(-2)=11,f(5)=-2, and f(x)=4.3`, what is the value of x?

Options

A. `-3.1`
B. `-1.9`
C. `1.6`
D. `2.9`

Explanation

To solve the problem, we will follow these steps:

1. **Identify the points given**: We have two points from the function:
- Point 1: \( (-2, 11) \)
- Point 2: \( (5, -2) \)

2. **Calculate the slope (m)**: The slope of a linear function can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, we can assign:
- \( x_1 = -2, y_1 = 11 \)
- \( x_2 = 5, y_2 = -2 \)

Substituting the values:
\[ m = \frac{-2 - 11}{5 - (-2)} = \frac{-13}{7} \]

3. **Use the slope to find the equation of the line**: The equation of a line in point-slope form is given by:
\[ y - y_1 = m(x - x_1) \]
Using point \( (-2, 11) \):
\[ y - 11 = \frac{-13}{7}(x + 2) \]

4. **Rearranging to find the equation in slope-intercept form**:
\[ y - 11 = \frac{-13}{7}x - \frac{26}{7} \]
Adding 11 to both sides:
\[ y = \frac{-13}{7}x + 11 - \frac{26}{7} \]
Converting 11 to a fraction:
\[ 11 = \frac{77}{7} \]
Thus:
\[ y = \frac{-13}{7}x + \frac{77}{7} - \frac{26}{7} = \frac{-13}{7}x + \frac{51}{7} \]

5. **Set the equation equal to 4.3**: We need to find \( x \) such that \( f(x) = 4.3 \):
\[ \frac{-13}{7}x + \frac{51}{7} = 4.3 \]

6. **Convert 4.3 to a fraction**:
\[ 4.3 = \frac{43}{10} \]

7. **Clear the fractions**: Multiply the entire equation by 70 (the least common multiple of 7 and 10):
\[ 70 \left(\frac{-13}{7}x\right) + 70 \left(\frac{51}{7}\right) = 70 \left(\frac{43}{10}\right) \]
This simplifies to:
\[ -130x + 510 = 301 \]

8. **Solve for \( x \)**:
\[ -130x = 301 - 510 \]
\[ -130x = -209 \]
\[ x = \frac{-209}{-130} = \frac{209}{130} \approx 1.6077 \]

9. **Final answer**: Rounding to one decimal place, we find:
\[ x \approx 1.6 \]