Question

The blue and orange lines represent a system. Use the sliders to manipulate the orange line to determine which equations would create a system that has no solution. Check all that apply. Hint: Convert to slope-intercept form.
2x + y = –5
y – 3 = 2x
x + y = –2
–2x = y
2x = 4 – y

Answer 1

To determine which equations create a system with no solution, we need to find equations that represent parallel lines. Parallel lines have the same slope but different y-intercepts. To do this, we will convert each equation to the slope-intercept form \(y = mx + b\), where \(m\) is the slope.

1. **\(2x + y = -5\)**
Solve for \(y\):
\[
y = -2x - 5
\]
Slope: -2

2. **\(y - 3 = 2x\)**
Solve for \(y\):
\[
y = 2x + 3
\]
Slope: 2

3. **\(x + y = -2\)**
Solve for \(y\):
\[
y = -x - 2
\]
Slope: -1

4. **\(-2x = y\)**
This is already in slope-intercept form:
\[
y = -2x
\]
Slope: -2

5. **\(2x = 4 - y\)**
Solve for \(y\):
\[
y = 4 - 2x
\]
Slope: -2

Now, we compare the slopes:

- The first equation \(y = -2x - 5\) has a slope of -2.
- The second equation \(y = 2x + 3\) has a slope of 2.
- The third equation \(y = -x - 2\) has a slope of -1.
- The fourth equation \(y = -2x\) has a slope of -2.
- The fifth equation \(y = 4 - 2x\) has a slope of -2.

The equations with a slope of -2 are:
- \(2x + y = -5\)
- \(-2x = y\)
- \(2x = 4 - y\)

Since these three equations have the same slope (-2) but different y-intercepts, they will be parallel to each other, leading to no solution.

Therefore, the equations that would create a system that has no solution are:
- \(2x + y = -5\)
- \(-2x = y\)
- \(2x = 4 - y\)

In summary, the equations with no solutions are:
- \(2x + y = -5\)
- \(-2x = y\)
- \(2x = 4 - y\)