Question
Solve the following system of equations graphically on the set of axes below.
y, equals, minus, x, plus, 4
y=−x+4
y, equals, 2, x, minus, 5
y=2x−5
Plot two lines by clicking the graph.
Click a line to delete it.
y, equals, minus, x, plus, 4
y=−x+4
y, equals, 2, x, minus, 5
y=2x−5
Plot two lines by clicking the graph.
Click a line to delete it.
Answers
GPT-4o mini
To solve the system of equations graphically, we will plot the two equations on the same set of axes.
1. **Equation 1:** \(y = -x + 4\)
- This line has a y-intercept of 4 (0, 4).
- The slope is -1, meaning for every unit you move right (increasing x), you move down 1 unit (decreasing y).
- Another point can be found by picking x = 2:
- \(y = -2 + 4 = 2\) → (2, 2)
Thus, we have two points for the line \(y = -x + 4\): (0, 4) and (2, 2).
2. **Equation 2:** \(y = 2x - 5\)
- This line has a y-intercept of -5 (0, -5).
- The slope is 2, meaning for every unit you move right, you move up 2 units.
- Picking x = 2:
- \(y = 2(2) - 5 = 4 - 5 = -1\) → (2, -1)
Thus, we have two points for the line \(y = 2x - 5\): (0, -5) and (2, -1).
### Graph Representation:
- **Line 1** connects (0, 4) and (2, 2).
- **Line 2** connects (0, -5) and (2, -1).
### Finding the Intersection:
To find the point of intersection of these two lines, set the equations equal to each other:
\[
-x + 4 = 2x - 5
\]
Combine like terms:
\[
4 + 5 = 3x
\]
\[
9 = 3x
\]
\[
x = 3
\]
Substituting \(x = 3\) back into either original equation (let’s use the first equation):
\[
y = -3 + 4 = 1
\]
### Conclusion:
The point of intersection, which is the solution to the system of equations, is:
\[
(3, 1)
\]
### Graphically:
You would click to plot the two lines based on the points we calculated, then identify the intersection point at (3, 1) as the solution to the system.
1. **Equation 1:** \(y = -x + 4\)
- This line has a y-intercept of 4 (0, 4).
- The slope is -1, meaning for every unit you move right (increasing x), you move down 1 unit (decreasing y).
- Another point can be found by picking x = 2:
- \(y = -2 + 4 = 2\) → (2, 2)
Thus, we have two points for the line \(y = -x + 4\): (0, 4) and (2, 2).
2. **Equation 2:** \(y = 2x - 5\)
- This line has a y-intercept of -5 (0, -5).
- The slope is 2, meaning for every unit you move right, you move up 2 units.
- Picking x = 2:
- \(y = 2(2) - 5 = 4 - 5 = -1\) → (2, -1)
Thus, we have two points for the line \(y = 2x - 5\): (0, -5) and (2, -1).
### Graph Representation:
- **Line 1** connects (0, 4) and (2, 2).
- **Line 2** connects (0, -5) and (2, -1).
### Finding the Intersection:
To find the point of intersection of these two lines, set the equations equal to each other:
\[
-x + 4 = 2x - 5
\]
Combine like terms:
\[
4 + 5 = 3x
\]
\[
9 = 3x
\]
\[
x = 3
\]
Substituting \(x = 3\) back into either original equation (let’s use the first equation):
\[
y = -3 + 4 = 1
\]
### Conclusion:
The point of intersection, which is the solution to the system of equations, is:
\[
(3, 1)
\]
### Graphically:
You would click to plot the two lines based on the points we calculated, then identify the intersection point at (3, 1) as the solution to the system.