Question
Solve the following system of equations graphically on the set of axes below.
y= -1/5x - 8
y= 3/5x - 4
Plot two lines by clicking the graph.
Click a line to delete it.
y= -1/5x - 8
y= 3/5x - 4
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 first need to plot the two lines represented by the equations:
1. **Equation 1:** \( y = -\frac{1}{5}x - 8 \)
2. **Equation 2:** \( y = \frac{3}{5}x - 4 \)
### Step 1: Plotting the First Line
For the equation \( y = -\frac{1}{5}x - 8 \):
- The y-intercept is at \( (0, -8) \).
- The slope is \(-\frac{1}{5}\), which means for every 5 units you move to the right (positive x-direction), you move 1 unit down (negative y-direction).
**Points to plot:**
- \( (0, -8) \)
- If \( x = 5 \): \( y = -\frac{1}{5}(5) - 8 = -1 - 8 = -9 \) → Point \( (5, -9) \)
- If \( x = -5 \): \( y = -\frac{1}{5}(-5) - 8 = 1 - 8 = -7 \) → Point \( (-5, -7) \)
### Step 2: Plotting the Second Line
For the equation \( y = \frac{3}{5}x - 4 \):
- The y-intercept is at \( (0, -4) \).
- The slope is \(\frac{3}{5}\), meaning for every 5 units you move to the right, you move 3 units up.
**Points to plot:**
- \( (0, -4) \)
- If \( x = 5 \): \( y = \frac{3}{5}(5) - 4 = 3 - 4 = -1 \) → Point \( (5, -1) \)
- If \( x = -5 \): \( y = \frac{3}{5}(-5) - 4 = -3 - 4 = -7 \) → Point \( (-5, -7) \)
### Step 3: Drawing the Lines
1. Plot the points for the first line: \( (0, -8) \) and \( (5, -9) \), then draw a line through those points.
2. Plot the points for the second line: \( (0, -4) \) and \( (5, -1) \), then draw a line through those points.
### Step 4: Finding the Intersection
The point where the two lines intersect is the solution to the system of equations. Graphically, look for where the two lines cross.
You can use a graphing tool or graph paper to accurately plot these points and lines, and find the intersection point.
### Conclusion
After plotting the lines, visually inspect where they meet to find the solution to the system of equations. If you want numerical precision, you can set the two equations equal to each other and solve for \( x \) and \( y \) algebraically.
Would you like help with the algebraic solution as well?
1. **Equation 1:** \( y = -\frac{1}{5}x - 8 \)
2. **Equation 2:** \( y = \frac{3}{5}x - 4 \)
### Step 1: Plotting the First Line
For the equation \( y = -\frac{1}{5}x - 8 \):
- The y-intercept is at \( (0, -8) \).
- The slope is \(-\frac{1}{5}\), which means for every 5 units you move to the right (positive x-direction), you move 1 unit down (negative y-direction).
**Points to plot:**
- \( (0, -8) \)
- If \( x = 5 \): \( y = -\frac{1}{5}(5) - 8 = -1 - 8 = -9 \) → Point \( (5, -9) \)
- If \( x = -5 \): \( y = -\frac{1}{5}(-5) - 8 = 1 - 8 = -7 \) → Point \( (-5, -7) \)
### Step 2: Plotting the Second Line
For the equation \( y = \frac{3}{5}x - 4 \):
- The y-intercept is at \( (0, -4) \).
- The slope is \(\frac{3}{5}\), meaning for every 5 units you move to the right, you move 3 units up.
**Points to plot:**
- \( (0, -4) \)
- If \( x = 5 \): \( y = \frac{3}{5}(5) - 4 = 3 - 4 = -1 \) → Point \( (5, -1) \)
- If \( x = -5 \): \( y = \frac{3}{5}(-5) - 4 = -3 - 4 = -7 \) → Point \( (-5, -7) \)
### Step 3: Drawing the Lines
1. Plot the points for the first line: \( (0, -8) \) and \( (5, -9) \), then draw a line through those points.
2. Plot the points for the second line: \( (0, -4) \) and \( (5, -1) \), then draw a line through those points.
### Step 4: Finding the Intersection
The point where the two lines intersect is the solution to the system of equations. Graphically, look for where the two lines cross.
You can use a graphing tool or graph paper to accurately plot these points and lines, and find the intersection point.
### Conclusion
After plotting the lines, visually inspect where they meet to find the solution to the system of equations. If you want numerical precision, you can set the two equations equal to each other and solve for \( x \) and \( y \) algebraically.
Would you like help with the algebraic solution as well?