Question

Match the system of equations to the number of solutions it has.
y = 5x + 7 3y - 15x = 18 no, infinite or one solution
y = 3x + 6 y = -13x - 4 no, infinite or one solution
x - 2y = 6 3x - 6y = 18 no, infinite or one solution
y = 23x - 1 y = 23x - 2 no, infinite or one solution

Answer 1

To determine the number of solutions for each system of equations, we can analyze the equations and see if they are parallel, identical, or intersecting lines.

1. **System**:
\(y = 5x + 7\)
\(3y - 15x = 18\)
- Rewrite the second equation:
\[
3y = 15x + 18 \implies y = 5x + 6
\]
- Since the first equation is \(y = 5x + 7\) and now we have \(y = 5x + 6\), they are parallel lines (same slope, different intercepts).
**Solution**: no solutions.

2. **System**:
\(y = 3x + 6\)
\(y = -13x - 4\)
- Both are in slope-intercept form with different slopes (3 and -13), so they will intersect at one point.
**Solution**: one solution.

3. **System**:
\(x - 2y = 6\)
\(3x - 6y = 18\)
- Rewrite the second equation:
\[
3x - 6y = 18 \implies x - 2y = 6
\]
- The second equation is a multiple of the first, so they represent the same line.
**Solution**: infinite solutions.

4. **System**:
\(y = 23x - 1\)
\(y = 23x - 2\)
- These lines have the same slope (23) but different y-intercepts (-1 and -2), which means they are parallel.
**Solution**: no solutions.

### Summary of Solutions:
1. \(y = 5x + 7\) and \(3y - 15x = 18\) - **no solutions**.
2. \(y = 3x + 6\) and \(y = -13x - 4\) - **one solution**.
3. \(x - 2y = 6\) and \(3x - 6y = 18\) - **infinite solutions**.
4. \(y = 23x - 1\) and \(y = 23x - 2\) - **no solutions**.