Question
Use the graph to answer the question
The graph shows a line passing through (1, 60) and (7, 140)
Write the equation of the line shown in the graph
The graph shows a line passing through (1, 60) and (7, 140)
Write the equation of the line shown in the graph
Answers
GPT-4o mini
To write the equation of the line that passes through the points (1, 60) and (7, 140), we first need to determine the slope (m) of the line using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Let \((x_1, y_1) = (1, 60)\) and \((x_2, y_2) = (7, 140)\).
Substituting in the coordinates:
\[
m = \frac{140 - 60}{7 - 1} = \frac{80}{6} = \frac{40}{3}
\]
Now that we have the slope \(m = \frac{40}{3}\), we can use the point-slope form of the equation of a line, which is:
\[
y - y_1 = m(x - x_1)
\]
Using the point (1, 60):
\[
y - 60 = \frac{40}{3}(x - 1)
\]
Now, we can distribute the slope \(\frac{40}{3}\):
\[
y - 60 = \frac{40}{3}x - \frac{40}{3}
\]
Now, add 60 to both sides:
\[
y = \frac{40}{3}x - \frac{40}{3} + 60
\]
To simplify \(60\) in terms of thirds:
\[
60 = \frac{180}{3}
\]
So:
\[
y = \frac{40}{3}x - \frac{40}{3} + \frac{180}{3}
\]
Combine the constants:
\[
y = \frac{40}{3}x + \frac{140}{3}
\]
Thus, the equation of the line is:
\[
y = \frac{40}{3}x + \frac{140}{3}
\]
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Let \((x_1, y_1) = (1, 60)\) and \((x_2, y_2) = (7, 140)\).
Substituting in the coordinates:
\[
m = \frac{140 - 60}{7 - 1} = \frac{80}{6} = \frac{40}{3}
\]
Now that we have the slope \(m = \frac{40}{3}\), we can use the point-slope form of the equation of a line, which is:
\[
y - y_1 = m(x - x_1)
\]
Using the point (1, 60):
\[
y - 60 = \frac{40}{3}(x - 1)
\]
Now, we can distribute the slope \(\frac{40}{3}\):
\[
y - 60 = \frac{40}{3}x - \frac{40}{3}
\]
Now, add 60 to both sides:
\[
y = \frac{40}{3}x - \frac{40}{3} + 60
\]
To simplify \(60\) in terms of thirds:
\[
60 = \frac{180}{3}
\]
So:
\[
y = \frac{40}{3}x - \frac{40}{3} + \frac{180}{3}
\]
Combine the constants:
\[
y = \frac{40}{3}x + \frac{140}{3}
\]
Thus, the equation of the line is:
\[
y = \frac{40}{3}x + \frac{140}{3}
\]