Question
Solve the following system of equations graphically on the set of axes below.
y, equals, x, minus, 7
y=x−7
y, equals, minus, one third, x, minus, 3
y=−
3
1
x−3
Plot two lines by clicking the graph.
Click a line to delete it.
Solution:
give detailed instructions on how to graph
y, equals, x, minus, 7
y=x−7
y, equals, minus, one third, x, minus, 3
y=−
3
1
x−3
Plot two lines by clicking the graph.
Click a line to delete it.
Solution:
give detailed instructions on how to graph
Answers
GPT-4o mini
To solve the system of equations graphically, we need to graph the equations on a set of axes. In this case, we have two equations to plot:
1. \( y = x - 7 \)
2. \( y = -\frac{1}{3}x - 3 \)
Here's a step-by-step guide on how to graph these lines manually (or using graphing software, if available):
### Step-by-Step Instructions
1. **Identify the equations**:
- **Line 1**: \( y = x - 7 \)
- **Line 2**: \( y = -\frac{1}{3}x - 3 \)
2. **Graphing Line 1: \( y = x - 7 \)**
- **Find the y-intercept**: Set \( x = 0 \) (to find where the line crosses the y-axis).
- \( y = 0 - 7 = -7 \)
- So, the y-intercept is (0, -7).
- **Find another point**: Choose another value for \( x \), say \( x = 7 \).
- \( y = 7 - 7 = 0 \)
- So, another point is (7, 0).
- **Plot the points on the graph**: Draw the points (0, -7) and (7, 0) on the graph.
- **Draw the line**: Connect the two points with a straight line, extending it across the graph.
3. **Graphing Line 2: \( y = -\frac{1}{3}x - 3 \)**
- **Find the y-intercept**: Again, set \( x = 0 \).
- \( y = -\frac{1}{3}(0) - 3 = -3 \)
- So, the y-intercept is (0, -3).
- **Find another point**: Choose another value for \( x \), let’s say \( x = 3 \).
- \( y = -\frac{1}{3}(3) - 3 = -1 - 3 = -4 \)
- So, another point is (3, -4).
- **Plot the points on the graph**: Draw the points (0, -3) and (3, -4) on the graph.
- **Draw the line**: Connect the two points with a straight line, extending it across the graph.
4. **Determine the intersection**: The solution to the system of equations is the point where the two lines intersect.
5. **Identify the intersection point**: Look for the point where the two lines cross on the graph. You may also choose to solve for the intersection algebraically if needed.
### Summary of the Intersection Point (Optional)
After plotting the lines, observe where they cross. You can also solve the system of equations simultaneously to confirm the intersection point:
To find it algebraically, set the right-hand sides equal:
\[
x - 7 = -\frac{1}{3}x - 3
\]
**Combine and solve for x**:
1. Get rid of the fractions by multiplying both sides by 3:
\[
3(x - 7) = -x - 9
\]
2. Distribute:
\[
3x - 21 = -x - 9
\]
3. Add \( x \) to both sides:
\[
3x + x - 21 = -9
\]
4. Combine like terms:
\[
4x - 21 = -9
\]
5. Add 21 to both sides:
\[
4x = 12
\]
6. Divide by 4:
\[
x = 3
\]
Now substitute \( x = 3 \) into one of the original equations to find \( y \):
\[
y = 3 - 7 = -4
\]
So, the intersection point is \( (3, -4) \).
### Conclusion
By following these instructions, you can graph the two lines and find their intersection point both graphically and algebraically. The solution to the system is \( (3, -4) \).
1. \( y = x - 7 \)
2. \( y = -\frac{1}{3}x - 3 \)
Here's a step-by-step guide on how to graph these lines manually (or using graphing software, if available):
### Step-by-Step Instructions
1. **Identify the equations**:
- **Line 1**: \( y = x - 7 \)
- **Line 2**: \( y = -\frac{1}{3}x - 3 \)
2. **Graphing Line 1: \( y = x - 7 \)**
- **Find the y-intercept**: Set \( x = 0 \) (to find where the line crosses the y-axis).
- \( y = 0 - 7 = -7 \)
- So, the y-intercept is (0, -7).
- **Find another point**: Choose another value for \( x \), say \( x = 7 \).
- \( y = 7 - 7 = 0 \)
- So, another point is (7, 0).
- **Plot the points on the graph**: Draw the points (0, -7) and (7, 0) on the graph.
- **Draw the line**: Connect the two points with a straight line, extending it across the graph.
3. **Graphing Line 2: \( y = -\frac{1}{3}x - 3 \)**
- **Find the y-intercept**: Again, set \( x = 0 \).
- \( y = -\frac{1}{3}(0) - 3 = -3 \)
- So, the y-intercept is (0, -3).
- **Find another point**: Choose another value for \( x \), let’s say \( x = 3 \).
- \( y = -\frac{1}{3}(3) - 3 = -1 - 3 = -4 \)
- So, another point is (3, -4).
- **Plot the points on the graph**: Draw the points (0, -3) and (3, -4) on the graph.
- **Draw the line**: Connect the two points with a straight line, extending it across the graph.
4. **Determine the intersection**: The solution to the system of equations is the point where the two lines intersect.
5. **Identify the intersection point**: Look for the point where the two lines cross on the graph. You may also choose to solve for the intersection algebraically if needed.
### Summary of the Intersection Point (Optional)
After plotting the lines, observe where they cross. You can also solve the system of equations simultaneously to confirm the intersection point:
To find it algebraically, set the right-hand sides equal:
\[
x - 7 = -\frac{1}{3}x - 3
\]
**Combine and solve for x**:
1. Get rid of the fractions by multiplying both sides by 3:
\[
3(x - 7) = -x - 9
\]
2. Distribute:
\[
3x - 21 = -x - 9
\]
3. Add \( x \) to both sides:
\[
3x + x - 21 = -9
\]
4. Combine like terms:
\[
4x - 21 = -9
\]
5. Add 21 to both sides:
\[
4x = 12
\]
6. Divide by 4:
\[
x = 3
\]
Now substitute \( x = 3 \) into one of the original equations to find \( y \):
\[
y = 3 - 7 = -4
\]
So, the intersection point is \( (3, -4) \).
### Conclusion
By following these instructions, you can graph the two lines and find their intersection point both graphically and algebraically. The solution to the system is \( (3, -4) \).