Question
Which of the following is the equation that represents the graph?
Graph of a line the passes through the points negative 6 comma 0 and 0 comma negative 4.
y equals negative two thirds times x minus 6
y equals negative three halves times x minus 6
y equals negative two thirds times x minus 4
y equals negative three halves times x minus 4
Graph of a line the passes through the points negative 6 comma 0 and 0 comma negative 4.
y equals negative two thirds times x minus 6
y equals negative three halves times x minus 6
y equals negative two thirds times x minus 4
y equals negative three halves times x minus 4
Answers
Answered by
GPT-4o mini
To determine the equation of the line that passes through the points (-6, 0) and (0, -4), we first need to find the slope of the line.
The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the points \((-6, 0)\) and \((0, -4)\):
\[
m = \frac{-4 - 0}{0 - (-6)} = \frac{-4}{0 + 6} = \frac{-4}{6} = -\frac{2}{3}
\]
Now that we have the slope \( m = -\frac{2}{3} \), we can use the point-slope form of the line equation:
\[
y - y_1 = m(x - x_1)
\]
We can choose one of the two points; let's use the point (0, -4):
\[
y - (-4) = -\frac{2}{3}(x - 0)
\]
This simplifies to:
\[
y + 4 = -\frac{2}{3}x
\]
Subtracting 4 from both sides gives:
\[
y = -\frac{2}{3}x - 4
\]
Thus, the correct equation of the line is:
\[
y = -\frac{2}{3}x - 4
\]
Now, let's match this with the options given:
1. \(y = -\frac{2}{3}(x - 6)\) -- This would not correctly represent the line.
2. \(y = -\frac{3}{2}(x - 6)\) -- This has the wrong slope.
3. \(y = -\frac{2}{3}(x - 4)\) -- This represents a different line.
4. \(y = -\frac{3}{2}(x - 4)\) -- This also has the wrong slope.
None of the provided options correctly represent the equation derived.
If you would like to use either point in the point-slope form again or check the options in different terms, or if you can provide more details about the adaptation of these options, I can help further.
The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the points \((-6, 0)\) and \((0, -4)\):
\[
m = \frac{-4 - 0}{0 - (-6)} = \frac{-4}{0 + 6} = \frac{-4}{6} = -\frac{2}{3}
\]
Now that we have the slope \( m = -\frac{2}{3} \), we can use the point-slope form of the line equation:
\[
y - y_1 = m(x - x_1)
\]
We can choose one of the two points; let's use the point (0, -4):
\[
y - (-4) = -\frac{2}{3}(x - 0)
\]
This simplifies to:
\[
y + 4 = -\frac{2}{3}x
\]
Subtracting 4 from both sides gives:
\[
y = -\frac{2}{3}x - 4
\]
Thus, the correct equation of the line is:
\[
y = -\frac{2}{3}x - 4
\]
Now, let's match this with the options given:
1. \(y = -\frac{2}{3}(x - 6)\) -- This would not correctly represent the line.
2. \(y = -\frac{3}{2}(x - 6)\) -- This has the wrong slope.
3. \(y = -\frac{2}{3}(x - 4)\) -- This represents a different line.
4. \(y = -\frac{3}{2}(x - 4)\) -- This also has the wrong slope.
None of the provided options correctly represent the equation derived.
If you would like to use either point in the point-slope form again or check the options in different terms, or if you can provide more details about the adaptation of these options, I can help further.