Question
Which graph shows the solution to the system of linear equations?
y equals one half times x
y equals negative one half times x minus 4
coordinate plane with one line that passes through the points 0 comma negative 4 and 2 comma negative 5 and another line that passes through the points 0 comma 0 and 2 comma 1
coordinate plane with one line that passes through the points 0 comma 2 and negative 3 comma 3 and another line that passes through the points 0 comma 0 and negative 3 comma negative 1
coordinate plane with one line that passes through the points 3 comma negative 3 and 0 comma negative 2 and another line that passes through the points 0 comma 0 and 3 comma 1
coordinate plane with one line that passes through the points 0 comma 4 and negative 1 comma 1 and another line that passes through the points 0 comma 0 and 1 comma 3
y equals one half times x
y equals negative one half times x minus 4
coordinate plane with one line that passes through the points 0 comma negative 4 and 2 comma negative 5 and another line that passes through the points 0 comma 0 and 2 comma 1
coordinate plane with one line that passes through the points 0 comma 2 and negative 3 comma 3 and another line that passes through the points 0 comma 0 and negative 3 comma negative 1
coordinate plane with one line that passes through the points 3 comma negative 3 and 0 comma negative 2 and another line that passes through the points 0 comma 0 and 3 comma 1
coordinate plane with one line that passes through the points 0 comma 4 and negative 1 comma 1 and another line that passes through the points 0 comma 0 and 1 comma 3
Answers
Answered by
GPT-4o mini
To determine which graph shows the solution to the given system of linear equations, we first need to analyze the two equations:
1. **Equation 1**: \( y = \frac{1}{2}x \)
This line passes through the origin (0, 0) with a slope of \( \frac{1}{2} \). Therefore, some points on this line include:
- (0, 0)
- (2, 1)
- (4, 2)
- etc.
2. **Equation 2**: \( y = -\frac{1}{2}x - 4 \)
This line has a y-intercept of -4 and a slope of \( -\frac{1}{2} \). It passes through:
- (0, -4) (y-intercept)
- (2, -5) (when x=2, y = -1 - 4 = -5)
- (-2, -3) (when x=-2, y = 1 - 4 = -3)
- etc.
Now, we will look at the options provided to find a graph that contains:
- The first line passing through (0, 0) and (2, 1).
- The second line passing through (0, -4) and (2, -5).
From the descriptions:
1. **Option 1**: Shows a line passing through (0, -4) and (2, -5) as well as a line that passes through (0, 0) and (2, 1). This matches both equations.
2. **Option 2**: Contains a line passing through (0, 0) and (-3, -1), which does not correspond to either equation.
3. **Option 3**: Has a line passing through (0, 0) and (3, 1), which does not correspond to the equations.
4. **Option 4**: Has a line through (0, 4) and (-1, 1), which does not correspond to the equations.
Thus, **Option 1** is the correct graph representing the system of linear equations.
1. **Equation 1**: \( y = \frac{1}{2}x \)
This line passes through the origin (0, 0) with a slope of \( \frac{1}{2} \). Therefore, some points on this line include:
- (0, 0)
- (2, 1)
- (4, 2)
- etc.
2. **Equation 2**: \( y = -\frac{1}{2}x - 4 \)
This line has a y-intercept of -4 and a slope of \( -\frac{1}{2} \). It passes through:
- (0, -4) (y-intercept)
- (2, -5) (when x=2, y = -1 - 4 = -5)
- (-2, -3) (when x=-2, y = 1 - 4 = -3)
- etc.
Now, we will look at the options provided to find a graph that contains:
- The first line passing through (0, 0) and (2, 1).
- The second line passing through (0, -4) and (2, -5).
From the descriptions:
1. **Option 1**: Shows a line passing through (0, -4) and (2, -5) as well as a line that passes through (0, 0) and (2, 1). This matches both equations.
2. **Option 2**: Contains a line passing through (0, 0) and (-3, -1), which does not correspond to either equation.
3. **Option 3**: Has a line passing through (0, 0) and (3, 1), which does not correspond to the equations.
4. **Option 4**: Has a line through (0, 4) and (-1, 1), which does not correspond to the equations.
Thus, **Option 1** is the correct graph representing the system of linear equations.