Question

On a coordinate plane, a line with negative slope goes through (0, 1) and (1, negative 1).
The graphed line is y = –2x + 1.

Which equation, when graphed with the given equation, will form a system that has no solution?
y = 2x – 3
y + 2x = 1
y = –2x – 3
y –2x = 1

Answer 1

To find an equation that will form a system with no solution when graphed alongside the given line \( y = -2x + 1 \), we need to find an equation that represents a line that is parallel to \( y = -2x + 1 \) but has a different y-intercept.

Lines that are parallel have the same slope. The slope of the given line \( y = -2x + 1 \) is \(-2\). Therefore, any line that is parallel to it will also have a slope of \(-2\), and for the system to have no solution, the y-intercept must be different.

Let's examine each of the provided options:

1. **\( y = 2x - 3 \)**:
- The slope is \(2\), which is not parallel to \(-2\). This line will intersect the given line, so it would have a solution.

2. **\( y + 2x = 1 \)**:
- Rewrite it in slope-intercept form: \( y = -2x + 1 \).
- This is the same line as the given line. Therefore, it has infinitely many solutions.

3. **\( y = -2x - 3 \)**:
- The slope is \(-2\), which is parallel to the given line. The y-intercept is \(-3\), which is different from \(1\). Hence, this line will never intersect with the given line, resulting in no solution.

4. **\( y - 2x = 1 \)**:
- Rewrite it in slope-intercept form: \( y = 2x + 1 \).
- The slope is \(2\) which is not parallel to \(-2\). Thus, this line will intersect the given line, leading to a solution.

The only equation that will form a system with no solution is:

**\( y = -2x - 3 \)**.