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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.

3 answers

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:

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

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)

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:

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

  1. 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:

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