Question
Question 5
(Multiple Choice Worth 2 Points)
(Systems of Linear Equations MC)
Solve the system of linear equations by graphing.
y equals negative one half times x plus 2
y equals one half times x minus 3
negative 5 comma negative one half
5 comma negative one half
negative one half comma 5
negative one half comma negative 5
Answers
GPT-4o mini
To solve the system of linear equations by graphing, we need to graph the two equations:
1. \( y = -\frac{1}{2}x + 2 \)
2. \( y = \frac{1}{2}x - 3 \)
**Step 1: Graph the first equation.**
For \( y = -\frac{1}{2}x + 2 \):
- The y-intercept is 2 (when \( x = 0 \), \( y = 2 \)).
- The slope is -1/2, which means for every 2 units you move horizontally to the right (positive x direction), you move down 1 unit (negative y direction).
Plotting two points:
- If \( x = 0 \), then \( y = 2 \) → Point (0, 2).
- If \( x = 2 \), then \( y = -1 \) → Point (2, -1) (since \( y = -\frac{1}{2}(2) + 2 = -1 \)).
**Step 2: Graph the second equation.**
For \( y = \frac{1}{2}x - 3 \):
- The y-intercept is -3 (when \( x = 0 \), \( y = -3 \)).
- The slope is 1/2, which means for every 2 units you move horizontally to the right, you move up 1 unit.
Plotting two points:
- If \( x = 0 \), then \( y = -3 \) → Point (0, -3).
- If \( x = 2 \), then \( y = -2 \) → Point (2, -2) (since \( y = \frac{1}{2}(2) - 3 = -2 \)).
**Step 3: Find the intersection point.**
Now, you would look for the point where the two lines intersect when graphed correctly. However, since this is a multiple-choice question, let's check the provided options to see which one might be the solution.
**Candidate Options:**
- \( (-5, -\frac{1}{2}) \)
- \( (5, -\frac{1}{2}) \)
- \( (-\frac{1}{2}, 5) \)
- \( (-\frac{1}{2}, -5) \)
To find the intersection a little more directly, we can set the two equations equal to each other:
\[
-\frac{1}{2}x + 2 = \frac{1}{2}x - 3
\]
Combining like terms:
\[
2 + 3 = \frac{1}{2}x + \frac{1}{2}x
\]
\[
5 = x
\]
Now substitute \( x = 5 \) back into one of the original equations to find \( y \):
Using the first equation:
\[
y = -\frac{1}{2}(5) + 2 = -\frac{5}{2} + 2 = -\frac{5}{2} + \frac{4}{2} = -\frac{1}{2}
\]
Thus, the intersection point of the two lines is \( (5, -\frac{1}{2}) \).
**Final Answer:**
- \( 5, -\frac{1}{2} \) is the solution to the system of equations.
1. \( y = -\frac{1}{2}x + 2 \)
2. \( y = \frac{1}{2}x - 3 \)
**Step 1: Graph the first equation.**
For \( y = -\frac{1}{2}x + 2 \):
- The y-intercept is 2 (when \( x = 0 \), \( y = 2 \)).
- The slope is -1/2, which means for every 2 units you move horizontally to the right (positive x direction), you move down 1 unit (negative y direction).
Plotting two points:
- If \( x = 0 \), then \( y = 2 \) → Point (0, 2).
- If \( x = 2 \), then \( y = -1 \) → Point (2, -1) (since \( y = -\frac{1}{2}(2) + 2 = -1 \)).
**Step 2: Graph the second equation.**
For \( y = \frac{1}{2}x - 3 \):
- The y-intercept is -3 (when \( x = 0 \), \( y = -3 \)).
- The slope is 1/2, which means for every 2 units you move horizontally to the right, you move up 1 unit.
Plotting two points:
- If \( x = 0 \), then \( y = -3 \) → Point (0, -3).
- If \( x = 2 \), then \( y = -2 \) → Point (2, -2) (since \( y = \frac{1}{2}(2) - 3 = -2 \)).
**Step 3: Find the intersection point.**
Now, you would look for the point where the two lines intersect when graphed correctly. However, since this is a multiple-choice question, let's check the provided options to see which one might be the solution.
**Candidate Options:**
- \( (-5, -\frac{1}{2}) \)
- \( (5, -\frac{1}{2}) \)
- \( (-\frac{1}{2}, 5) \)
- \( (-\frac{1}{2}, -5) \)
To find the intersection a little more directly, we can set the two equations equal to each other:
\[
-\frac{1}{2}x + 2 = \frac{1}{2}x - 3
\]
Combining like terms:
\[
2 + 3 = \frac{1}{2}x + \frac{1}{2}x
\]
\[
5 = x
\]
Now substitute \( x = 5 \) back into one of the original equations to find \( y \):
Using the first equation:
\[
y = -\frac{1}{2}(5) + 2 = -\frac{5}{2} + 2 = -\frac{5}{2} + \frac{4}{2} = -\frac{1}{2}
\]
Thus, the intersection point of the two lines is \( (5, -\frac{1}{2}) \).
**Final Answer:**
- \( 5, -\frac{1}{2} \) is the solution to the system of equations.