Question
Which graph represents the solution set to this system of equations?
–x + 2y = 6 and 4x + y = 3
On a coordinate plane, a line goes through (negative 1, negative 1) and (0, 3) and another line goes through (1, 4) and (2, 2).
On a coordinate plane, a line goes through (negative 4, 0) and (0, 4) and another line goes through (negative 1, 1) and (0, negative 3).
On a coordinate plane, a line goes through (0, 3) and (2, 4) and another line goes through (0, 3) and (1, 0).
On a coordinate plane, a line goes through (0, 0) and (4, 3) and another line goes through (2, 5) and (4, 4).
–x + 2y = 6 and 4x + y = 3
On a coordinate plane, a line goes through (negative 1, negative 1) and (0, 3) and another line goes through (1, 4) and (2, 2).
On a coordinate plane, a line goes through (negative 4, 0) and (0, 4) and another line goes through (negative 1, 1) and (0, negative 3).
On a coordinate plane, a line goes through (0, 3) and (2, 4) and another line goes through (0, 3) and (1, 0).
On a coordinate plane, a line goes through (0, 0) and (4, 3) and another line goes through (2, 5) and (4, 4).
Answers
GPT-4o mini
To find which graph represents the solution set to the system of equations –x + 2y = 6 and 4x + y = 3, we can first solve the equations to find the intersection point.
1. **Solve the first equation**:
\[
-x + 2y = 6 \quad \Rightarrow \quad 2y = x + 6 \quad \Rightarrow \quad y = \frac{1}{2}x + 3
\]
This line has a y-intercept of 3 and a slope of 1/2.
2. **Solve the second equation**:
\[
4x + y = 3 \quad \Rightarrow \quad y = -4x + 3
\]
This line has a y-intercept of 3 and a slope of -4.
3. **Find the intersection point of the two lines** by setting their equations equal to each other:
\[
\frac{1}{2}x + 3 = -4x + 3
\]
\[
\frac{1}{2}x = -4x
\]
\[
\frac{1}{2}x + 4x = 0
\]
\[
\frac{9}{2}x = 0 \quad \Rightarrow \quad x = 0
\]
Substitute \(x = 0\) into one of the original equations to find \(y\):
\[
y = -4(0) + 3 = 3
\]
Thus, the intersection point is \((0, 3)\).
**Conclusion**: The solution set for the system of equations is the point where the two lines intersect, which is at (0, 3).
Now, let’s analyze the given options to find which graph has the point (0, 3) as an intersection:
- **Option 1**: Lines through (−1, −1) and (0, 3) & (1, 4) and (2, 2) – Contains (0, 3).
- **Option 2**: Lines through (−4, 0) and (0, 4) & (−1, 1) and (0, −3) – Contains (0, 4), not (0, 3).
- **Option 3**: Lines through (0, 3) and (2, 4) & (0, 3) and (1, 0) – Contains (0, 3).
- **Option 4**: Lines through (0, 0) and (4, 3) & (2, 5) and (4, 4) – Neither line contains (0, 3).
**Final Answer**: The graphs representing the lines that match the equations are found in options 1 and 3. However, since both options have indicated the point (0, 3), we would choose one of them; either works in terms of representing the system of equations correctly.
If you need a distinct answer, option 3 is more straightforwardly confirming point (0, 3) visually:
Option 3 is the final correct choice.
1. **Solve the first equation**:
\[
-x + 2y = 6 \quad \Rightarrow \quad 2y = x + 6 \quad \Rightarrow \quad y = \frac{1}{2}x + 3
\]
This line has a y-intercept of 3 and a slope of 1/2.
2. **Solve the second equation**:
\[
4x + y = 3 \quad \Rightarrow \quad y = -4x + 3
\]
This line has a y-intercept of 3 and a slope of -4.
3. **Find the intersection point of the two lines** by setting their equations equal to each other:
\[
\frac{1}{2}x + 3 = -4x + 3
\]
\[
\frac{1}{2}x = -4x
\]
\[
\frac{1}{2}x + 4x = 0
\]
\[
\frac{9}{2}x = 0 \quad \Rightarrow \quad x = 0
\]
Substitute \(x = 0\) into one of the original equations to find \(y\):
\[
y = -4(0) + 3 = 3
\]
Thus, the intersection point is \((0, 3)\).
**Conclusion**: The solution set for the system of equations is the point where the two lines intersect, which is at (0, 3).
Now, let’s analyze the given options to find which graph has the point (0, 3) as an intersection:
- **Option 1**: Lines through (−1, −1) and (0, 3) & (1, 4) and (2, 2) – Contains (0, 3).
- **Option 2**: Lines through (−4, 0) and (0, 4) & (−1, 1) and (0, −3) – Contains (0, 4), not (0, 3).
- **Option 3**: Lines through (0, 3) and (2, 4) & (0, 3) and (1, 0) – Contains (0, 3).
- **Option 4**: Lines through (0, 0) and (4, 3) & (2, 5) and (4, 4) – Neither line contains (0, 3).
**Final Answer**: The graphs representing the lines that match the equations are found in options 1 and 3. However, since both options have indicated the point (0, 3), we would choose one of them; either works in terms of representing the system of equations correctly.
If you need a distinct answer, option 3 is more straightforwardly confirming point (0, 3) visually:
Option 3 is the final correct choice.