Question

Solve the following system of equations graphically on the set of axes below.
y, equals, minus, 2, x, minus, 4
y=−2x−4
x, minus, y, equals, 7
x−y=7

Answer 1

x−y=7

Answer 2

To solve the system of equations graphically, we first need to rewrite both equations in slope-intercept form (y = mx + b) if they are not already in that form.

1. The first equation is already in slope-intercept form:

\[
y = -2x - 4
\]

2. The second equation can be rewritten as follows:

\[
x - y = 7 \implies y = x - 7
\]

Now we have the two equations:
1. \( y = -2x - 4 \)
2. \( y = x - 7 \)

Next, we will determine the points where these lines intersect by graphing them.

### Graphing the equations:

1. **Graphing \( y = -2x - 4 \)**:
- The y-intercept is -4. So the line crosses the y-axis at (0, -4).
- The slope is -2, which means for every increase of 1 in \( x \), \( y \) decreases by 2.
- When \( x = 1 \): \( y = -2(1) - 4 = -6 \) → Point (1, -6).
- When \( x = -1 \): \( y = -2(-1) - 4 = -2 \) → Point (-1, -2).

2. **Graphing \( y = x - 7 \)**:
- The y-intercept is -7. So the line crosses the y-axis at (0, -7).
- The slope is 1, meaning for every increase of 1 in \( x \), \( y \) increases by 1.
- When \( x = 1 \): \( y = 1 - 7 = -6 \) → Point (1, -6).
- When \( x = -2 \): \( y = -2 - 7 = -9 \) → Point (-2, -9).

### Finding the intersection:

To find the point where the two lines intersect, we can set the equations equal to each other:

\[
-2x - 4 = x - 7
\]

Now, we solve for \( x \):

1. Add \( 2x \) to both sides:
\[
-4 = 3x - 7
\]

2. Add 7 to both sides:
\[
3 = 3x
\]

3. Divide by 3:
\[
x = 1
\]

Now, substitute \( x = 1 \) back into either of the original equations to find \( y \):

Using \( y = x - 7 \):
\[
y = 1 - 7 = -6
\]

So the intersection point is \( (1, -6) \).

### Conclusion:

The graphical solution of the system of equations yields the point of intersection, which is:

\[
\boxed{(1, -6)}
\]

This point is the solution to the system of equations.

Answer 3

To analyze the equation \(x - y = 7\), we can isolate \(y\) to rewrite it in slope-intercept form. Here are the steps to do that:

### Step to Isolate \(y\)

Start with the equation:
\[
x - y = 7
\]

Now, isolate \(y\):
\[
-y = 7 - x
\]

Multiply through by -1 to solve for \(y\):
\[
y = x - 7
\]

### Summary of the Equations

You have two equations:

1. **From your previous message**:
\[
y = -2x - 4
\]

2. **From your current message**:
\[
y = x - 7
\]

### Finding the Intersection Graphically

To solve this graphically, we can plot both equations and find their intersection.

1. **Graphing \(y = -2x - 4\)**:
- **Y-intercept**: At \(x = 0, y = -4\) → point (0, -4).
- **Another point**: At \(x = 1\), \(y = -2(1) - 4 = -6\) → point (1, -6).

2. **Graphing \(y = x - 7\)**:
- **Y-intercept**: At \(x = 0, y = -7\) → point (0, -7).
- **Another point**: At \(x = 1\), \(y = 1 - 7 = -6\) → point (1, -6).

### Finding the Intersection Point Algebraically

To find the intersection, set the equations equal to each other:

\[
-2x - 4 = x - 7
\]

Solve for \(x\):

1. Add \(2x\) to both sides:
\[
-4 = 3x - 7
\]

2. Add 7 to both sides:
\[
3 = 3x
\]

3. Divide by 3:
\[
x = 1
\]

Now substitute \(x = 1\) back into one of the original equations to find \(y\):

Using \(y = x - 7\):
\[
y = 1 - 7 = -6
\]

### Conclusion

The intersection point (the solution to the system of equations) is:
\[
\boxed{(1, -6)}
\]

Thus, both equations intersect at the point \((1, -6)\).