Question
Which graph represents this system?
y = 2 x + 1. y = negative 4 x + 7.
On a coordinate plane, a line goes through (0, 7) and (1, 3) and another goes through (1, 3) and (2, 5).
On a coordinate plane, a line goes through (negative 4, 0) and (0, 7) and another line goes through (negative 4, 3) and (0, 1).
On a coordinate plane, a line goes through (0, negative 4) and (1, 3) and another goes through (0, 2) and (1, 3).
On a coordinate plane, a line goes through (0, 7) and (8, 5) and another line goes through (0, 1) and (8, 5).
y = 2 x + 1. y = negative 4 x + 7.
On a coordinate plane, a line goes through (0, 7) and (1, 3) and another goes through (1, 3) and (2, 5).
On a coordinate plane, a line goes through (negative 4, 0) and (0, 7) and another line goes through (negative 4, 3) and (0, 1).
On a coordinate plane, a line goes through (0, negative 4) and (1, 3) and another goes through (0, 2) and (1, 3).
On a coordinate plane, a line goes through (0, 7) and (8, 5) and another line goes through (0, 1) and (8, 5).
Answers
GPT-4o mini
To determine which graph represents the system of equations given by \( y = 2x + 1 \) and \( y = -4x + 7 \), we can analyze both equations:
1. **Equation 1: \( y = 2x + 1 \)**
- This is a linear equation in slope-intercept form \( y = mx + b \), where \( m = 2 \) (the slope) and \( b = 1 \) (the y-intercept).
- The line will intersect the y-axis at (0, 1) and has a positive slope, meaning it increases as \( x \) increases. It will also pass through points such as:
- If \( x = 0 \), \( y = 1 \) → (0, 1)
- If \( x = 1 \), \( y = 2(1) + 1 = 3 \) → (1, 3)
- If \( x = 2 \), \( y = 2(2) + 1 = 5 \) → (2, 5)
2. **Equation 2: \( y = -4x + 7 \)**
- This is also in slope-intercept form where \( m = -4 \) and \( b = 7 \).
- The line will intersect the y-axis at (0, 7) and has a negative slope, meaning it decreases as \( x \) increases. It will pass through points such as:
- If \( x = 0 \), \( y = 7 \) → (0, 7)
- If \( x = 1 \), \( y = -4(1) + 7 = 3 \) → (1, 3)
- If \( x = 2 \), \( y = -4(2) + 7 = -1 \) → (2, -1)
### Now let's analyze the graph options provided:
1. **First Option**: A line goes through (0, 7) and (1, 3) and another goes through (1, 3) and (2, 5).
- The line through (0, 7) to (1, 3) looks like it would correspond to \( y = -4x + 7 \).
- However, the second line through (1, 3) and (2, 5) does not correspond with \( y = 2x + 1 \), as it should be passing through (0, 1).
2. **Second Option**: A line goes through (-4, 0) and (0, 7) and another line goes through (-4, 3) and (0, 1).
- The line from (-4, 0) to (0, 7) could represent \( y = -4x + 7 \) definitely, but the other line (which doesn't correspond to our equations) doesn't match either.
3. **Third Option**: A line goes through (0, -4) and (1, 3) and another goes through (0, 2) and (1, 3).
- The lines do not match either of our equations.
4. **Fourth Option**: A line goes through (0, 7) and (8, 5) and another line goes through (0, 1) and (8, 5).
- The first line from (0, 7) would represent \( y = -4x + 7 \) and the second line does not correspond to \( y = 2x + 1 \).
### Conclusion
Upon carefully analyzing all options, none directly depicts the lines defined by the equations \( y = 2x + 1 \) and \( y = -4x + 7 \) correctly. However, the first line \( y = -4x + 7 \) could be closely approximated in **the first option** since it passes through (0, 7) and (1, 3), but the second part (line through (1, 3) and (2, 5)) is incorrect. Therefore, if asked to guess which graph represents the closest, it would be first but the option is incorrect due to the corresponding second line being wrong. No graph fully satisfies both equations.
1. **Equation 1: \( y = 2x + 1 \)**
- This is a linear equation in slope-intercept form \( y = mx + b \), where \( m = 2 \) (the slope) and \( b = 1 \) (the y-intercept).
- The line will intersect the y-axis at (0, 1) and has a positive slope, meaning it increases as \( x \) increases. It will also pass through points such as:
- If \( x = 0 \), \( y = 1 \) → (0, 1)
- If \( x = 1 \), \( y = 2(1) + 1 = 3 \) → (1, 3)
- If \( x = 2 \), \( y = 2(2) + 1 = 5 \) → (2, 5)
2. **Equation 2: \( y = -4x + 7 \)**
- This is also in slope-intercept form where \( m = -4 \) and \( b = 7 \).
- The line will intersect the y-axis at (0, 7) and has a negative slope, meaning it decreases as \( x \) increases. It will pass through points such as:
- If \( x = 0 \), \( y = 7 \) → (0, 7)
- If \( x = 1 \), \( y = -4(1) + 7 = 3 \) → (1, 3)
- If \( x = 2 \), \( y = -4(2) + 7 = -1 \) → (2, -1)
### Now let's analyze the graph options provided:
1. **First Option**: A line goes through (0, 7) and (1, 3) and another goes through (1, 3) and (2, 5).
- The line through (0, 7) to (1, 3) looks like it would correspond to \( y = -4x + 7 \).
- However, the second line through (1, 3) and (2, 5) does not correspond with \( y = 2x + 1 \), as it should be passing through (0, 1).
2. **Second Option**: A line goes through (-4, 0) and (0, 7) and another line goes through (-4, 3) and (0, 1).
- The line from (-4, 0) to (0, 7) could represent \( y = -4x + 7 \) definitely, but the other line (which doesn't correspond to our equations) doesn't match either.
3. **Third Option**: A line goes through (0, -4) and (1, 3) and another goes through (0, 2) and (1, 3).
- The lines do not match either of our equations.
4. **Fourth Option**: A line goes through (0, 7) and (8, 5) and another line goes through (0, 1) and (8, 5).
- The first line from (0, 7) would represent \( y = -4x + 7 \) and the second line does not correspond to \( y = 2x + 1 \).
### Conclusion
Upon carefully analyzing all options, none directly depicts the lines defined by the equations \( y = 2x + 1 \) and \( y = -4x + 7 \) correctly. However, the first line \( y = -4x + 7 \) could be closely approximated in **the first option** since it passes through (0, 7) and (1, 3), but the second part (line through (1, 3) and (2, 5)) is incorrect. Therefore, if asked to guess which graph represents the closest, it would be first but the option is incorrect due to the corresponding second line being wrong. No graph fully satisfies both equations.