Question
The graph below represents a system of equations.
A line passing through (-2, 0) and (0, -3), another line passing through (0, 1) and (2, 0) intersect at (-4, 3).
Which of the following statements is true?
A.
The system has infinite solutions.
B.
The solutions to the system are
and
.
C.
The solution to the system is
.
D.
The system has no solution.
A line passing through (-2, 0) and (0, -3), another line passing through (0, 1) and (2, 0) intersect at (-4, 3).
Which of the following statements is true?
A.
The system has infinite solutions.
B.
The solutions to the system are
and
.
C.
The solution to the system is
.
D.
The system has no solution.
Answers
GPT-4o mini
To determine the correct statement based on the information provided about the system of equations, let's analyze the situation further.
1. **Identifying the lines:**
- The first line passes through the points (-2, 0) and (0, -3). We can calculate its slope and equation:
- The slope (m) is calculated as:
\[
m = \frac{-3 - 0}{0 - (-2)} = \frac{-3}{2}
\]
- Using the point-slope form (y - y₁ = m(x - x₁)) with the point (0, -3):
\[
y + 3 = -\frac{3}{2}(x - 0)
\]
Simplifying gives:
\[
y = -\frac{3}{2}x - 3
\]
- The second line passes through the points (0, 1) and (2, 0):
- The slope (m) is:
\[
m = \frac{0 - 1}{2 - 0} = \frac{-1}{2}
\]
- Using the point-slope form with the point (0, 1):
\[
y - 1 = -\frac{1}{2}(x - 0)
\]
Simplifying gives:
\[
y = -\frac{1}{2}x + 1
\]
2. **Finding the intersection:**
- According to your description, these two lines intersect at (-4, 3). To confirm this is the solution of the system of equations, we check if (-4, 3) satisfies both equations.
- For the first line \(y = -\frac{3}{2}x - 3\):
\[
3 = -\frac{3}{2}(-4) - 3
\]
\[
3 = 6 - 3 \quad \text{(True)}
\]
- For the second line \(y = -\frac{1}{2}x + 1\):
\[
3 = -\frac{1}{2}(-4) + 1
\]
\[
3 = 2 + 1 \quad \text{(True)}
\]
3. **Interpreting the results:**
- Both lines intersect at the point (-4, 3). Therefore, this point is the only solution to the system of equations.
Given this analysis, we can evaluate the statements:
A. **The system has infinite solutions.** ➔ False (They intersect at one point).
B. **The solutions to the system are and .** ➔ Incomplete.
C. **The solution to the system is .** ➔ True (but not 100% complete without specifying the coordinates).
D. **The system has no solution.** ➔ False (there is a point of intersection).
The most accurate statement among the provided options is **C**, indicating that the solution to the system is the point where the two lines intersect, which is (-4, 3).
1. **Identifying the lines:**
- The first line passes through the points (-2, 0) and (0, -3). We can calculate its slope and equation:
- The slope (m) is calculated as:
\[
m = \frac{-3 - 0}{0 - (-2)} = \frac{-3}{2}
\]
- Using the point-slope form (y - y₁ = m(x - x₁)) with the point (0, -3):
\[
y + 3 = -\frac{3}{2}(x - 0)
\]
Simplifying gives:
\[
y = -\frac{3}{2}x - 3
\]
- The second line passes through the points (0, 1) and (2, 0):
- The slope (m) is:
\[
m = \frac{0 - 1}{2 - 0} = \frac{-1}{2}
\]
- Using the point-slope form with the point (0, 1):
\[
y - 1 = -\frac{1}{2}(x - 0)
\]
Simplifying gives:
\[
y = -\frac{1}{2}x + 1
\]
2. **Finding the intersection:**
- According to your description, these two lines intersect at (-4, 3). To confirm this is the solution of the system of equations, we check if (-4, 3) satisfies both equations.
- For the first line \(y = -\frac{3}{2}x - 3\):
\[
3 = -\frac{3}{2}(-4) - 3
\]
\[
3 = 6 - 3 \quad \text{(True)}
\]
- For the second line \(y = -\frac{1}{2}x + 1\):
\[
3 = -\frac{1}{2}(-4) + 1
\]
\[
3 = 2 + 1 \quad \text{(True)}
\]
3. **Interpreting the results:**
- Both lines intersect at the point (-4, 3). Therefore, this point is the only solution to the system of equations.
Given this analysis, we can evaluate the statements:
A. **The system has infinite solutions.** ➔ False (They intersect at one point).
B. **The solutions to the system are and .** ➔ Incomplete.
C. **The solution to the system is .** ➔ True (but not 100% complete without specifying the coordinates).
D. **The system has no solution.** ➔ False (there is a point of intersection).
The most accurate statement among the provided options is **C**, indicating that the solution to the system is the point where the two lines intersect, which is (-4, 3).