Question

Complete steps through in order to determine which pair of equations form a system with one solution or infinite solutions
Determine which equations are in slope intercept form if not then solve the equation for y to identify the slope intercept form
1- y=5x-4
2- y=-2x+3
3- y=3x+1
4- 2y=6x+2
5- y-5x=+2
6- y+4x+1=0

Answer 1

To determine which pair of equations form a system with one solution or infinite solutions, we need to first express all equations in slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Then, we can analyze the slopes of the equations.

**Step 1: Convert equations to slope-intercept form.**

1. **\(y = 5x - 4\)**
Already in slope-intercept form.
Slope (m) = 5, Intercept (b) = -4

2. **\(y = -2x + 3\)**
Already in slope-intercept form.
Slope (m) = -2, Intercept (b) = 3

3. **\(y = 3x + 1\)**
Already in slope-intercept form.
Slope (m) = 3, Intercept (b) = 1

4. **\(2y = 6x + 2\)**
Divide by 2 to isolate \(y\):
\(y = 3x + 1\)
Now in slope-intercept form.
Slope (m) = 3, Intercept (b) = 1

5. **\(y - 5x = 2\)**
Rearranging gives:
\(y = 5x + 2\)
Now in slope-intercept form.
Slope (m) = 5, Intercept (b) = 2

6. **\(y + 4x + 1 = 0\)**
Rearranging gives:
\(y = -4x - 1\)
Now in slope-intercept form.
Slope (m) = -4, Intercept (b) = -1

**Step 2: Summary of the equations in slope-intercept form**
- 1. \(y = 5x - 4\) (m = 5)
- 2. \(y = -2x + 3\) (m = -2)
- 3. \(y = 3x + 1\) (m = 3)
- 4. \(y = 3x + 1\) (m = 3)
- 5. \(y = 5x + 2\) (m = 5)
- 6. \(y = -4x - 1\) (m = -4)

**Step 3: Determine pairs of equations**

1. **Equations 1 & 5:**
\(m_1 = 5\), \(m_5 = 5\) (same slope, different intercept) ⟶ Infinite solutions (same line)

2. **Equations 2 & 6:**
\(m_2 = -2\), \(m_6 = -4\) (different slopes) ⟶ One solution

3. **Equations 3 & 4:**
\(m_3 = 3\), \(m_4 = 3\) (same slope, same intercept) ⟶ Infinite solutions (same line)

4. **Equations 1 & 4:**
\(m_1 = 5\), \(m_4 = 3\) (different slopes) ⟶ One solution

5. **Equations 2 & 3:**
\(m_2 = -2\), \(m_3 = 3\) (different slopes) ⟶ One solution

6. **Equations 5 & 6:**
\(m_5 = 5\), \(m_6 = -4\) (different slopes) ⟶ One solution

**Conclusion:**
- **Infinite Solutions:** Equations 1 & 5, Equations 3 & 4
- **One Solution:** All other pairs (2 & 6, 1 & 4, 2 & 3, 5 & 6)