Solve the following system of equations graphically on the set of axes below.
y, equals, x, plus, 6
y=x+6
y, equals, minus, start fraction, 3, divided by, 2, end fraction, x, minus, 4
y=−
2
3
​
x−4
Plot two lines by clicking the graph.
Click a line to delete it.
x
y
x
y
Answer
Attempt 1 out of 2

Solution:  Submit Answer

To solve the system of equations graphically, you need to plot both equations and find the point where they intersect, as this point represents the solution to the system.

### Equations to plot:
1. \( y = x + 6 \)
2. \( y = -\frac{3}{2}x - 4 \)

### Step-by-step process:

1. **Plot the first line \( y = x + 6 \):**
   - The slope of the line is \(1\) (rise/run = 1/1).
   - The y-intercept is 6, so the line passes through the point \( (0, 6) \).
   - Choose another point: if \( x = 1 \), then \( y = 1 + 6 = 7 \), giving you the point \( (1, 7) \).
   - Draw a line through the points \( (0, 6) \) and \( (1, 7) \).

2. **Plot the second line \( y = -\frac{3}{2}x - 4 \):**
   - The slope of the line is \( -\frac{3}{2} \) (rise/run = -3/2).
   - The y-intercept is -4, so the line passes through the point \( (0, -4) \).
   - Choose another point: if \( x = 2 \), then \( y = -\frac{3}{2}(2) - 4 = -3 - 4 = -7 \), giving you the point \( (2, -7) \).
   - Draw a line through the points \( (0, -4) \) and \( (2, -7) \).

3. **Find the intersection:**
   - The point where both lines intersect is the solution to the system of equations.

#### Let's solve the system algebraically as well to confirm the graphical result:
1. Set the two equations equal to each other:
   \[
   x + 6 = -\frac{3}{2}x - 4
   \]
2. Solve for \( x \):
   \[
   x + \frac{3}{2}x = -4 - 6
   \]
   \[
   \frac{5}{2}x = -10
   \]
   \[
   x = -4
   \]
3. Substitute \( x = -4 \) into the first equation:
   \[
   y = -4 + 6 = 2
   \]

So, the solution is \( x = -4 \) and \( y = 2 \), or the point \( (-4, 2) \).

### Conclusion:
The solution to the system of equations is \( (-4, 2) \). You can confirm this by plotting the two lines and checking where they intersect.

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Would you like more details on this solution or any other clarification? 

Here are 5 related questions to explore:
1. How do you graph a line given its slope and y-intercept?
2. What is the algebraic method of solving a system of linear equations?
3. How do you interpret the solution to a system of equations graphically?
4. What are different ways to solve a system of equations (besides graphing)?
5. How does changing the slope of a line affect its graph?

**Tip:** When solving graphically, always double-check the slope and y-intercept values to ensure accuracy in plotting.

Solve the following system of equations graphically on the set of axes below. y = x + 6 and y = -(3/2)x - 4. Plot two lines by clicking the graph. Find the solution where they intersect.

Learn how to graph the system of linear equations y = x + 6 and y = -(3/2)x - 4 and find the solution graphically. This guide walks you through the step-by-step process of plotting the lines and identifying the point of intersection, covering key concepts in graphing and systems of equations. Suitable for Grades 9-10.

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