Question
A line passes through the points (–3, 7) and (6, 4). Which shows the graph of this line?
On a coordinate plane, a line goes through points (0, 6) and (6, 4).
On a coordinate plane, a line goes through points (4, 6) and (6, 0).
On a coordinate plane, a line goes through points (2, negative 10) and (6, 4).
On a coordinate plane, a line goes through points (negative 10, 2) and (4, 6).
On a coordinate plane, a line goes through points (0, 6) and (6, 4).
On a coordinate plane, a line goes through points (4, 6) and (6, 0).
On a coordinate plane, a line goes through points (2, negative 10) and (6, 4).
On a coordinate plane, a line goes through points (negative 10, 2) and (4, 6).
Answers
GPT-4o mini
To find the equation of the line that passes through the points \((-3, 7)\) and \( (6, 4)\), we can first calculate the slope of the line using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using \((-3, 7)\) as \((x_1, y_1)\) and \((6, 4)\) as \((x_2, y_2)\):
\[
m = \frac{4 - 7}{6 - (-3)} = \frac{-3}{9} = -\frac{1}{3}
\]
Next, we can use point-slope form to write the equation of the line. Point-slope form is given by:
\[
y - y_1 = m(x - x_1)
\]
Using the point \((-3, 7)\) and the slope \(m = -\frac{1}{3}\):
\[
y - 7 = -\frac{1}{3}(x + 3)
\]
Expanding this, we get:
\[
y - 7 = -\frac{1}{3}x - 1
\]
Adding 7 to both sides gives:
\[
y = -\frac{1}{3}x + 6
\]
Now that we have the equation of the line, \(y = -\frac{1}{3}x + 6\), we can check its y-intercept and x-intercept:
- The y-intercept occurs when \(x = 0\):
\[
y = -\frac{1}{3}(0) + 6 = 6
\]
- The x-intercept occurs when \(y = 0\):
\[
0 = -\frac{1}{3}x + 6 \implies \frac{1}{3}x = 6 \implies x = 18
\]
With the slope of \(-\frac{1}{3}\), we see that the line is decreasing.
Now we can analyze the options presented in the question:
1. Line through points (0, 6) and (6, 4) — This intersects y=6 at x=0, which is consistent with our line.
2. Line through points (4, 6) and (6, 0) — These points do not correspond to the calculated slope.
3. Line through points (2, -10) and (6, 4) — These points diverge in slope from our line.
4. Line through points (-10, 2) and (4, 6) — Again, these points diverge in slope from our line.
**Conclusion**: The correct representation of the line is the first option, which goes through points (0, 6) and (6, 4).
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using \((-3, 7)\) as \((x_1, y_1)\) and \((6, 4)\) as \((x_2, y_2)\):
\[
m = \frac{4 - 7}{6 - (-3)} = \frac{-3}{9} = -\frac{1}{3}
\]
Next, we can use point-slope form to write the equation of the line. Point-slope form is given by:
\[
y - y_1 = m(x - x_1)
\]
Using the point \((-3, 7)\) and the slope \(m = -\frac{1}{3}\):
\[
y - 7 = -\frac{1}{3}(x + 3)
\]
Expanding this, we get:
\[
y - 7 = -\frac{1}{3}x - 1
\]
Adding 7 to both sides gives:
\[
y = -\frac{1}{3}x + 6
\]
Now that we have the equation of the line, \(y = -\frac{1}{3}x + 6\), we can check its y-intercept and x-intercept:
- The y-intercept occurs when \(x = 0\):
\[
y = -\frac{1}{3}(0) + 6 = 6
\]
- The x-intercept occurs when \(y = 0\):
\[
0 = -\frac{1}{3}x + 6 \implies \frac{1}{3}x = 6 \implies x = 18
\]
With the slope of \(-\frac{1}{3}\), we see that the line is decreasing.
Now we can analyze the options presented in the question:
1. Line through points (0, 6) and (6, 4) — This intersects y=6 at x=0, which is consistent with our line.
2. Line through points (4, 6) and (6, 0) — These points do not correspond to the calculated slope.
3. Line through points (2, -10) and (6, 4) — These points diverge in slope from our line.
4. Line through points (-10, 2) and (4, 6) — Again, these points diverge in slope from our line.
**Conclusion**: The correct representation of the line is the first option, which goes through points (0, 6) and (6, 4).