To find the equation of a linear function that represents the relationship between \( x \) and \( y \), we need to determine the slope \( m \) and the y-intercept \( b \) of the linear equation in the form:
\[ y = mx + b \]
Step 1: Calculate the Slope (m)
We can find the slope \( m \) using any two points from the table. Let's use the points (-4, 10) and (-2, 7):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 10}{-2 - (-4)} = \frac{-3}{2} = -\frac{3}{2} \]
Step 2: Calculate the Y-Intercept (b)
Now that we have the slope, we can use one of the points to find the y-intercept \( b \). Using the point (-4, 10):
\[ y = mx + b \] \[ 10 = -\frac{3}{2}(-4) + b \] \[ 10 = 6 + b \] \[ b = 10 - 6 = 4 \]
Step 3: Write the Equation
Now we can put the values of \( m \) and \( b \) together:
\[ y = -\frac{3}{2}x + 4 \]
Final Equation
So, the equation that could represent the relationship between \( x \) and \( y \) is:
\[ y = -\frac{3}{2}x + 4 \]
Filling in the Boxes
If you have specific options to choose from for \( m \) and \( b \), you can fill them in based on this equation:
- For the slope (m), it will be \(-\frac{3}{2}\)
- For the y-intercept (b), it will be \(4\)
Thus, the final answer should be arranged as:
y = -\frac{3}{2} x + 4
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