Question
The graph shows the system of equations below.
2 x minus 3 y = negative 6. y = negative one-third x minus 4
mc018-2.jpg
Which statement is supported by the graph?
The blue line represents the equation mc018-3.jpg, and the solution to the system of equations is (–6, –2).
The blue line represents the equation mc018-4.jpg, and the solution to the system of equations is ( –2, –6).
The red line represents the equation mc018-5.jpg, and the solution to the system of equations is (2, –4).
The red line represents the equation mc018-6.jpg, and the solution to the system of equations is (–4, 2).
2 x minus 3 y = negative 6. y = negative one-third x minus 4
mc018-2.jpg
Which statement is supported by the graph?
The blue line represents the equation mc018-3.jpg, and the solution to the system of equations is (–6, –2).
The blue line represents the equation mc018-4.jpg, and the solution to the system of equations is ( –2, –6).
The red line represents the equation mc018-5.jpg, and the solution to the system of equations is (2, –4).
The red line represents the equation mc018-6.jpg, and the solution to the system of equations is (–4, 2).
Answers
GPT-4o mini
To determine which statements are supported by the graph, we need to identify the two lines represented by the given equations and their intersection point (the solution).
The two equations are:
1. \(2x - 3y = -6\)
2. \(y = -\frac{1}{3}x - 4\)
**Step 1: Convert the first equation to slope-intercept form.**
Start with \(2x - 3y = -6\):
1. \( -3y = -2x - 6\)
2. \( y = \frac{2}{3}x + 2\)
Now we have two lines:
1. \(y = \frac{2}{3}x + 2\) (this corresponds to the red line)
2. \(y = -\frac{1}{3}x - 4\) (this corresponds to the blue line)
**Step 2: Identify the intersection point.**
To find the intersection point, set the two equations equal to each other:
\[
\frac{2}{3}x + 2 = -\frac{1}{3}x - 4
\]
Multiply each term by 3 to eliminate the fractions:
\[
2x + 6 = -x - 12
\]
Combine like terms:
\[
2x + x = -12 - 6
\]
\[
3x = -18
\]
\[
x = -6
\]
Now substitute \(x = -6\) into one of the equations to find \(y\):
Use \(y = -\frac{1}{3}(-6) - 4\):
\[
y = 2 - 4 = -2
\]
So, the intersection point (solution to the system of equations) is \((-6, -2)\).
**Step 3: Evaluate the statements.**
- **The blue line represents the equation \(y = -\frac{1}{3}x - 4\), and the solution to the system of equations is (-6, -2).** (Correct)
- The other options do not correctly match the equations or intersection point.
Therefore, the correct statement supported by the graph is:
**The blue line represents the equation \(y = -\frac{1}{3}x - 4\), and the solution to the system of equations is (-6, -2).**
The two equations are:
1. \(2x - 3y = -6\)
2. \(y = -\frac{1}{3}x - 4\)
**Step 1: Convert the first equation to slope-intercept form.**
Start with \(2x - 3y = -6\):
1. \( -3y = -2x - 6\)
2. \( y = \frac{2}{3}x + 2\)
Now we have two lines:
1. \(y = \frac{2}{3}x + 2\) (this corresponds to the red line)
2. \(y = -\frac{1}{3}x - 4\) (this corresponds to the blue line)
**Step 2: Identify the intersection point.**
To find the intersection point, set the two equations equal to each other:
\[
\frac{2}{3}x + 2 = -\frac{1}{3}x - 4
\]
Multiply each term by 3 to eliminate the fractions:
\[
2x + 6 = -x - 12
\]
Combine like terms:
\[
2x + x = -12 - 6
\]
\[
3x = -18
\]
\[
x = -6
\]
Now substitute \(x = -6\) into one of the equations to find \(y\):
Use \(y = -\frac{1}{3}(-6) - 4\):
\[
y = 2 - 4 = -2
\]
So, the intersection point (solution to the system of equations) is \((-6, -2)\).
**Step 3: Evaluate the statements.**
- **The blue line represents the equation \(y = -\frac{1}{3}x - 4\), and the solution to the system of equations is (-6, -2).** (Correct)
- The other options do not correctly match the equations or intersection point.
Therefore, the correct statement supported by the graph is:
**The blue line represents the equation \(y = -\frac{1}{3}x - 4\), and the solution to the system of equations is (-6, -2).**