Question
Solve the following system of equations graphically on the set of axes below.
y, equals, minus, start fraction, 3, divided by, 2, end fraction, x, minus, 6
y=−
2
3
x−6
y, equals, x, minus, 1
y=x−1
give detailed instructions on how to graph
y, equals, minus, start fraction, 3, divided by, 2, end fraction, x, minus, 6
y=−
2
3
x−6
y, equals, x, minus, 1
y=x−1
give detailed instructions on how to graph
Answers
GPT-4o mini
To solve the given system of equations graphically, you can follow these steps to graph each equation and find their point of intersection.
### Equations to Graph:
1. \( y = -\frac{3}{2}x - 6 \)
2. \( y = x - 1 \)
### Steps for Graphing:
#### Step 1: Set Up Your Axes
- Draw a horizontal line representing the y-axis and a vertical line representing the x-axis. Label your axes appropriately, and make sure to mark equal intervals for both axes.
#### Step 2: Graph the First Equation \( y = -\frac{3}{2}x - 6 \)
1. **Plot the y-intercept:**
- The y-intercept is where the graph crosses the y-axis, which occurs at \( -6 \). Plot the point (0, -6) on the graph.
2. **Use the slope to find another point:**
- The slope of this line is \( -\frac{3}{2} \), which means for every 2 units you move to the right (positive x-direction), you move down 3 units (negative y-direction).
- Starting from (0, -6):
- Move right 2 units to x = 2, and down 3 units:
- This results in the point \( (2, -9) \). Plot this point.
3. **Draw the line:**
- Draw a straight line through the points (0, -6) and (2, -9). Extend the line in both directions.
#### Step 3: Graph the Second Equation \( y = x - 1 \)
1. **Plot the y-intercept:**
- The y-intercept for this equation is \( -1 \). Plot the point (0, -1).
2. **Use the slope to find another point:**
- The slope of this line is \( 1 \), indicating that for every 1 unit you move to the right, you move up 1 unit.
- From (0, -1):
- Move right 1 unit to x = 1, and up 1 unit:
- This results in the point \( (1, 0) \). Plot this point.
3. **Draw the line:**
- Connect the points (0, -1) and (1, 0) with a straight line, extending it in both directions.
#### Step 4: Identify the Point of Intersection
- Look at your graph to see where the two lines intersect. This point represents the solution to the system of equations.
- Estimate the coordinates of the intersection. If the intersection is not clear, you can use substitution or elimination methods to find the exact coordinates.
### Example of Possible Intersection
After graphing, you might find that the lines intersect at a certain point, say \( (x, y) \). You can calculate this intersection precisely by either solving the equations algebraically or reading off from the graph.
### Conclusion
By following these steps, you'll be able to graph the given equations and identify the point where they intersect, which will be the solution to the system of equations.
### Equations to Graph:
1. \( y = -\frac{3}{2}x - 6 \)
2. \( y = x - 1 \)
### Steps for Graphing:
#### Step 1: Set Up Your Axes
- Draw a horizontal line representing the y-axis and a vertical line representing the x-axis. Label your axes appropriately, and make sure to mark equal intervals for both axes.
#### Step 2: Graph the First Equation \( y = -\frac{3}{2}x - 6 \)
1. **Plot the y-intercept:**
- The y-intercept is where the graph crosses the y-axis, which occurs at \( -6 \). Plot the point (0, -6) on the graph.
2. **Use the slope to find another point:**
- The slope of this line is \( -\frac{3}{2} \), which means for every 2 units you move to the right (positive x-direction), you move down 3 units (negative y-direction).
- Starting from (0, -6):
- Move right 2 units to x = 2, and down 3 units:
- This results in the point \( (2, -9) \). Plot this point.
3. **Draw the line:**
- Draw a straight line through the points (0, -6) and (2, -9). Extend the line in both directions.
#### Step 3: Graph the Second Equation \( y = x - 1 \)
1. **Plot the y-intercept:**
- The y-intercept for this equation is \( -1 \). Plot the point (0, -1).
2. **Use the slope to find another point:**
- The slope of this line is \( 1 \), indicating that for every 1 unit you move to the right, you move up 1 unit.
- From (0, -1):
- Move right 1 unit to x = 1, and up 1 unit:
- This results in the point \( (1, 0) \). Plot this point.
3. **Draw the line:**
- Connect the points (0, -1) and (1, 0) with a straight line, extending it in both directions.
#### Step 4: Identify the Point of Intersection
- Look at your graph to see where the two lines intersect. This point represents the solution to the system of equations.
- Estimate the coordinates of the intersection. If the intersection is not clear, you can use substitution or elimination methods to find the exact coordinates.
### Example of Possible Intersection
After graphing, you might find that the lines intersect at a certain point, say \( (x, y) \). You can calculate this intersection precisely by either solving the equations algebraically or reading off from the graph.
### Conclusion
By following these steps, you'll be able to graph the given equations and identify the point where they intersect, which will be the solution to the system of equations.