Question
Which graph represents this system?
3 x + 2 y = negative 6. y = negative three-halves x + 2.
On a coordinate plane, a line goes through (negative 3, negative 4) and (0, negative 6) and another line goes through (0, 2) and (3, 0).
On a coordinate plane, a line goes through (negative 2, 0) and (0, negative 3) and another goes through (0, 2) and (4, negative 4).
On a coordinate plane, a line goes through (0, 2) and (4, negative 4) and another goes through (0, negative 3) and (6, 1).
On a coordinate plane, a line goes through (0, 2) and (4, negative 4) and another goes through (0, negative 6) and (9, 0).
3 x + 2 y = negative 6. y = negative three-halves x + 2.
On a coordinate plane, a line goes through (negative 3, negative 4) and (0, negative 6) and another line goes through (0, 2) and (3, 0).
On a coordinate plane, a line goes through (negative 2, 0) and (0, negative 3) and another goes through (0, 2) and (4, negative 4).
On a coordinate plane, a line goes through (0, 2) and (4, negative 4) and another goes through (0, negative 3) and (6, 1).
On a coordinate plane, a line goes through (0, 2) and (4, negative 4) and another goes through (0, negative 6) and (9, 0).
Answers
GPT-4o mini
To determine which graph represents the system of equations `3x + 2y = -6` and `y = -3/2x + 2`, we need to analyze both equations and find their intercepts and slopes.
1. **First equation: \(3x + 2y = -6\)**
Rearranging to slope-intercept form (\(y = mx + b\)):
\[
2y = -3x - 6 \quad \Rightarrow \quad y = -\frac{3}{2}x - 3
\]
- The slope (m) is \(-\frac{3}{2}\).
- The y-intercept (b) is \(-3\) (point (0, -3)).
2. **Second equation: \(y = -\frac{3}{2}x + 2\)**
- The slope (m) is \(-\frac{3}{2}\).
- The y-intercept (b) is \(2\) (point (0, 2)).
Both lines have the same slope of \(-\frac{3}{2}\), meaning they are parallel. However, since they have different y-intercepts, they will not intersect and thus do not represent a unique system of equations.
Now, we examine the given graph descriptions for matching points:
- **Line 1** (from the first equation) goes through (0, -3), corresponding to its y-intercept.
- **Line 2** (from the second equation) goes through (0, 2), corresponding to its y-intercept.
From the available options, the graph that shows these characteristics:
1. Must have a point at (0, -3) for the first line.
2. Must have a point at (0, 2) for the second line.
Looking at your options:
- The graph option that describes both lines:
- A line going through (0, 2) and another going through (0, -3).
Therefore, the correct graph is likely **the one showing a line through (0, 2) and (4, -4)**, and **a line through (0, -3) and (6, 1)**.
This matches the system:
- One line crosses y-axis at (0, -3), and another at (0, 2), maintaining the required characteristics that we derived from the equations.
Thus, the graph can be determined from the description provided.
1. **First equation: \(3x + 2y = -6\)**
Rearranging to slope-intercept form (\(y = mx + b\)):
\[
2y = -3x - 6 \quad \Rightarrow \quad y = -\frac{3}{2}x - 3
\]
- The slope (m) is \(-\frac{3}{2}\).
- The y-intercept (b) is \(-3\) (point (0, -3)).
2. **Second equation: \(y = -\frac{3}{2}x + 2\)**
- The slope (m) is \(-\frac{3}{2}\).
- The y-intercept (b) is \(2\) (point (0, 2)).
Both lines have the same slope of \(-\frac{3}{2}\), meaning they are parallel. However, since they have different y-intercepts, they will not intersect and thus do not represent a unique system of equations.
Now, we examine the given graph descriptions for matching points:
- **Line 1** (from the first equation) goes through (0, -3), corresponding to its y-intercept.
- **Line 2** (from the second equation) goes through (0, 2), corresponding to its y-intercept.
From the available options, the graph that shows these characteristics:
1. Must have a point at (0, -3) for the first line.
2. Must have a point at (0, 2) for the second line.
Looking at your options:
- The graph option that describes both lines:
- A line going through (0, 2) and another going through (0, -3).
Therefore, the correct graph is likely **the one showing a line through (0, 2) and (4, -4)**, and **a line through (0, -3) and (6, 1)**.
This matches the system:
- One line crosses y-axis at (0, -3), and another at (0, 2), maintaining the required characteristics that we derived from the equations.
Thus, the graph can be determined from the description provided.