Question

x y
11 | 58
20|103 Write the equation of the linear function that models the relationship shown in the table. Enter your answer in y=mx+b form.

Answer 1

Write the equation of the linear function shown on the graph. Enter your answer in y=mx+b form. (1 point) (2,4) (-3,-1)

Answer 2

To find the equation of the linear function that models the relationship between x and y given the points (11, 58) and (20, 103), we first need to determine the slope (m) of the line. The slope is calculated using the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Here, we can assign:
- \( (x_1, y_1) = (11, 58) \)
- \( (x_2, y_2) = (20, 103) \)

Now, substituting the values into the slope formula:

\[
m = \frac{103 - 58}{20 - 11} = \frac{45}{9} = 5
\]

Now that we have the slope \( m = 5 \), we can use the point-slope form of a linear equation to find the y-intercept (b). We can use either of the two points; let's use the point (11, 58):

The point-slope form is:

\[
y - y_1 = m(x - x_1)
\]

Substituting the values into this equation:

\[
y - 58 = 5(x - 11)
\]

Now we solve for \(y\):

\[
y - 58 = 5x - 55
\]
\[
y = 5x - 55 + 58
\]
\[
y = 5x + 3
\]

Thus, the equation of the linear function in \(y = mx + b\) form is:

\[
\boxed{y = 5x + 3}
\]

Answer 3

To determine the equation of the linear function that passes through the points \((2, 4)\) and \((-3, -1)\), we need to find the slope (m) and then use one of the points to find the y-intercept (b).

1. **Calculate the slope (m)** using the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Here, we can assign:
- \( (x_1, y_1) = (2, 4) \)
- \( (x_2, y_2) = (-3, -1) \)

Substituting these values into the slope formula:

\[
m = \frac{-1 - 4}{-3 - 2} = \frac{-5}{-5} = 1
\]

2. **Now that we have the slope \(m = 1\)**, we can use the point-slope form to find the equation of the line. Using point \((2, 4)\):

The point-slope form is:

\[
y - y_1 = m(x - x_1)
\]

Substituting the known values:

\[
y - 4 = 1(x - 2)
\]

Now, we can simplify and solve for \(y\):

\[
y - 4 = x - 2
\]
\[
y = x - 2 + 4
\]
\[
y = x + 2
\]

Thus, the equation of the linear function in \(y = mx + b\) form is:

\[
\boxed{y = x + 2}
\]