Question

Match the system of equations to the number of solutions it has. (4 points)

y = 5x + 7 and 3y−15x=18

y = \frac{2}{3}x - 1 and y = \frac{2}{3}x - 2

x - 2y = 6 and 3x−6y=18

3x+6=y and y=−
3
1
​
x−4

No solutions One Solution Infinite solutions

Answer 1

Let's analyze each system of equations:

1. **System**: \( y = 5x + 7 \) and \( 3y - 15x = 18 \)

- The first equation is in slope-intercept form.
- The second equation can be rewritten as \( y = 5x + 6 \) by substituting \(3y = 15x + 18\) and then dividing by 3.
- Since both equations have the same slope (5), but different y-intercepts (7 and 6), they are parallel lines.
- **Number of solutions**: No solutions.

2. **System**: \( y = \frac{2}{3}x - 1 \) and \( y = \frac{2}{3}x - 2 \)

- Both equations are in slope-intercept form.
- They have the same slope (\(\frac{2}{3}\)) but different y-intercepts (-1 and -2).
- Hence, these lines are also parallel and will never intersect.
- **Number of solutions**: No solutions.

3. **System**: \( x - 2y = 6 \) and \( 3x - 6y = 18 \)

- The first equation can be rewritten as \( y = \frac{1}{2}x - 3 \).
- The second equation can be simplified to \( y = \frac{1}{2}x - 3 \).
- Both equations represent the same line since they are equivalent.
- **Number of solutions**: Infinite solutions.

4. **System**: \( 3x + 6 = y \) and \( y = -\frac{3}{1}x - 4 \)

- The first equation can be rewritten as \( y = 3x + 6 \).
- The second equation simplifies to \( y = -3x - 4 \).
- These lines have different slopes (3 and -3) and will intersect at exactly one point.
- **Number of solutions**: One solution.

**Summary:**
- **y = 5x + 7 and 3y - 15x = 18**: No solutions
- **y = \frac{2}{3}x - 1 and y = \frac{2}{3}x - 2**: No solutions
- **x - 2y = 6 and 3x - 6y = 18**: Infinite solutions
- **3x + 6 = y and y = -\frac{3}{1}x - 4**: One solution