Question

Graph these equations:
x–y=4
x+y=

6
Click to select points on the graph.

Answers

GPT-4o mini
To graph the equations \(x - y = 4\) and \(x + y = -6\), we can begin by rewriting them in slope-intercept form (y = mx + b).

1. **For the first equation**: \(x - y = 4\)
- Rearranging gives:
\[-y = -x + 4\]
- Multiplying through by -1 gives:
\[y = x - 4\]

This tells us that the slope (m) is 1, and the y-intercept (b) is -4. The line crosses the y-axis at (0, -4).

2. **For the second equation**: \(x + y = -6\)
- Rearranging gives:
\[y = -x - 6\]

Here, the slope (m) is -1, and the y-intercept (b) is -6. The line crosses the y-axis at (0, -6).

**Finding points for each line:**

- For **\(y = x - 4\)**:
- If \(x = 0\), then \(y = 0 - 4 = -4\) → point (0, -4)
- If \(x = 4\), then \(y = 4 - 4 = 0\) → point (4, 0)

- For **\(y = -x - 6\)**:
- If \(x = 0\), then \(y = -0 - 6 = -6\) → point (0, -6)
- If \(x = 4\), then \(y = -4 - 6 = -10\) → point (4, -10)

### Graphing the Lines:

1. Plot the points for the equation \(y = x - 4\): (0, -4) and (4, 0) and draw the line through them.
2. Plot the points for the equation \(y = -x - 6\): (0, -6) and (4, -10) and draw the line through them.

### Intersection:
The two lines will intersect at some point; you can find it by solving the system of equations.

To find the intersection:
1. Set \(x - 4 = -x - 6\).
2. Solve for \(x\):
\[
x + x = -6 + 4 \implies 2x = -2 \implies x = -1
\]
3. Substitute \(x = -1\) back into one of the original equations to find \(y\):
\[
y = -1 - 4 = -5
\]

Thus, the lines intersect at the point **(-1, -5)**.

### Summary of Points:
- From the first equation: (0, -4) and (4, 0)
- From the second equation: (0, -6) and (4, -10)
- Intersection point: (-1, -5)

You can now graph these points to visualize the lines!