Question
Which graph represents this system?
y = one-half x + 3. y = three-halves x minus 1
On a coordinate plane, a line goes through (0, 3) and (4, 5) and another goes through (0, negative 1) and (2, 2).
On a coordinate plane, a line goes through (0, 3) and (1, negative 3) and another goes through (0, negative 1) and (3, 1).
On a coordinate plane, a line goes through (negative 1, negative 2) and (1, 4) and another goes through (0, 1.5) and (1.5, 0).
On a coordinate plane, a line goes through (negative 3, negative 3) and (0, 3) and another goes through (0, negative 1) and (3, 1).
y = one-half x + 3. y = three-halves x minus 1
On a coordinate plane, a line goes through (0, 3) and (4, 5) and another goes through (0, negative 1) and (2, 2).
On a coordinate plane, a line goes through (0, 3) and (1, negative 3) and another goes through (0, negative 1) and (3, 1).
On a coordinate plane, a line goes through (negative 1, negative 2) and (1, 4) and another goes through (0, 1.5) and (1.5, 0).
On a coordinate plane, a line goes through (negative 3, negative 3) and (0, 3) and another goes through (0, negative 1) and (3, 1).
Answers
GPT-4o mini
To determine which graph represents the given system of equations, we need to analyze the equations and match them with the graphs provided.
The equations are:
1. **y = (1/2)x + 3**
2. **y = (3/2)x - 1**
Let's find the intercepts and slopes for both equations:
1. **For y = (1/2)x + 3**:
- Y-intercept: When x = 0, y = 3 (point (0, 3))
- Slope: 1/2 means that for every 2 units you move right (positive x-direction), you move 1 unit up.
2. **For y = (3/2)x - 1**:
- Y-intercept: When x = 0, y = -1 (point (0, -1))
- Slope: 3/2 means that for every 2 units you move right (positive x-direction), you move 3 units up.
Now let's check the coordinates of the lines:
- The first line should go through (0, 3) and somewhere that matches with the slope of 1/2.
- The second line should go through (0, -1) and should go up faster, matching the slope of 3/2.
Now, let's analyze the graph options:
1. **First graph**:
- One line through (0, 3) and (4, 5) (slope = 0.5).
- Another line through (0, -1) and (2, 2) (slope = 3/2).
- This graph matches both equations.
2. **Second graph**:
- One line through (0, 3) and (1, -3) (wrong slope).
- Another through (0, -1) and (3, 1) (wrong slope).
3. **Third graph**:
- Line through (-1, -2) and (1, 4) (wrong intercepts).
- Another through (0, 1.5) and (1.5, 0) (incorrect).
4. **Fourth graph**:
- Line through (-3, -3) and (0, 3) (not matching the right points).
- Another through (0, -1) and (3, 1) (wrong slope).
After checking all options, **the first graph correctly represents the given system** of equations.
The equations are:
1. **y = (1/2)x + 3**
2. **y = (3/2)x - 1**
Let's find the intercepts and slopes for both equations:
1. **For y = (1/2)x + 3**:
- Y-intercept: When x = 0, y = 3 (point (0, 3))
- Slope: 1/2 means that for every 2 units you move right (positive x-direction), you move 1 unit up.
2. **For y = (3/2)x - 1**:
- Y-intercept: When x = 0, y = -1 (point (0, -1))
- Slope: 3/2 means that for every 2 units you move right (positive x-direction), you move 3 units up.
Now let's check the coordinates of the lines:
- The first line should go through (0, 3) and somewhere that matches with the slope of 1/2.
- The second line should go through (0, -1) and should go up faster, matching the slope of 3/2.
Now, let's analyze the graph options:
1. **First graph**:
- One line through (0, 3) and (4, 5) (slope = 0.5).
- Another line through (0, -1) and (2, 2) (slope = 3/2).
- This graph matches both equations.
2. **Second graph**:
- One line through (0, 3) and (1, -3) (wrong slope).
- Another through (0, -1) and (3, 1) (wrong slope).
3. **Third graph**:
- Line through (-1, -2) and (1, 4) (wrong intercepts).
- Another through (0, 1.5) and (1.5, 0) (incorrect).
4. **Fourth graph**:
- Line through (-3, -3) and (0, 3) (not matching the right points).
- Another through (0, -1) and (3, 1) (wrong slope).
After checking all options, **the first graph correctly represents the given system** of equations.