Question
Which graph represents this system?
y = 3. x + y = 4.
On a coordinate plane, a horizontal line is at y = 3 and another line goes through (0, 4) and (4, 0).
On a coordinate plane, a vertical line is at x = 3 and another line goes through (0, 4) and (4, 0).
On a coordinate plane, a horizontal line is at y = 3 and another line goes through (0, 4) and (2, 6).
On a coordinate plane, a vertical line is at x = 3 and another line goes through (0, 4) and (3, 7).
y = 3. x + y = 4.
On a coordinate plane, a horizontal line is at y = 3 and another line goes through (0, 4) and (4, 0).
On a coordinate plane, a vertical line is at x = 3 and another line goes through (0, 4) and (4, 0).
On a coordinate plane, a horizontal line is at y = 3 and another line goes through (0, 4) and (2, 6).
On a coordinate plane, a vertical line is at x = 3 and another line goes through (0, 4) and (3, 7).
Answers
GPT-4o mini
To determine which graph represents the given system of equations, we first analyze both equations:
1. \( y = 3 \) is a horizontal line at \( y = 3 \).
2. The equation \( x + y = 4 \) can be rewritten in slope-intercept form (y = mx + b) as \( y = -x + 4 \).
Now, we can find the y-intercept and the x-intercept for the line \( x + y = 4 \):
- **Y-Intercept**: Set \( x = 0 \) → \( y = 4 \), so the point is (0, 4).
- **X-Intercept**: Set \( y = 0 \) → \( x = 4 \), so the point is (4, 0).
Therefore, the line \( x + y = 4 \) passes through the points (0, 4) and (4, 0) and has a negative slope.
Now, let's examine the provided options:
1. **Option 1**: A horizontal line at \( y = 3 \) and a line through (0, 4) and (4, 0).
- This matches our equations.
2. **Option 2**: A vertical line at \( x = 3 \) and a line through (0, 4) and (4, 0).
- The vertical line does not match \( x + y = 4 \).
3. **Option 3**: A horizontal line at \( y = 3 \) and a line through (0, 4) and (2, 6).
- The line through (0, 4) and (2, 6) has a positive slope, which does not fit \( x + y = 4 \).
4. **Option 4**: A vertical line at \( x = 3 \) and a line through (0, 4) and (3, 7).
- Again, the vertical line does not match \( x + y = 4 \), and the line does not have the correct slope.
Thus, the only option that accurately represents the system of equations is:
**Option 1**: A horizontal line at \( y = 3 \) and a line through (0, 4) and (4, 0).
1. \( y = 3 \) is a horizontal line at \( y = 3 \).
2. The equation \( x + y = 4 \) can be rewritten in slope-intercept form (y = mx + b) as \( y = -x + 4 \).
Now, we can find the y-intercept and the x-intercept for the line \( x + y = 4 \):
- **Y-Intercept**: Set \( x = 0 \) → \( y = 4 \), so the point is (0, 4).
- **X-Intercept**: Set \( y = 0 \) → \( x = 4 \), so the point is (4, 0).
Therefore, the line \( x + y = 4 \) passes through the points (0, 4) and (4, 0) and has a negative slope.
Now, let's examine the provided options:
1. **Option 1**: A horizontal line at \( y = 3 \) and a line through (0, 4) and (4, 0).
- This matches our equations.
2. **Option 2**: A vertical line at \( x = 3 \) and a line through (0, 4) and (4, 0).
- The vertical line does not match \( x + y = 4 \).
3. **Option 3**: A horizontal line at \( y = 3 \) and a line through (0, 4) and (2, 6).
- The line through (0, 4) and (2, 6) has a positive slope, which does not fit \( x + y = 4 \).
4. **Option 4**: A vertical line at \( x = 3 \) and a line through (0, 4) and (3, 7).
- Again, the vertical line does not match \( x + y = 4 \), and the line does not have the correct slope.
Thus, the only option that accurately represents the system of equations is:
**Option 1**: A horizontal line at \( y = 3 \) and a line through (0, 4) and (4, 0).