Question

Graphing Linear Equations Portfolio
Worksheet
Directions: Use this worksheet to record your answers to the graphing linear
equations activities in this portfolio. When you are finished, save this worksheet
with your answers and submit it for a portfolio grade.
Investigate and explain the different types of graphical solutions.
Question 1
Examine the different equations that follow:
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
7. y=5x
Complete steps a. through e. to determine which pair of equations form a system
with one solution, no solution, or infinite solutions.
a. Solve each equation and make sure it is in y=mx+b format.
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
7. y

Answer 1

Sure! Let's go through each equation and convert them into the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

### Step a: Rewrite the Equations in \(y = mx + b\) Format

1. **\(y = 5x - 4\)**
- This equation is already in slope-intercept form.

2. **\(y = -2x + 3\)**
- This equation is also already in slope-intercept form.

3. **\(y = 3x + 1\)**
- This equation is already in slope-intercept form.

4. **\(2y = 6x + 2\)**
- To convert to \(y = mx + b\):
\[
y = \frac{6}{2}x + \frac{2}{2} = 3x + 1
\]

5. **\(y - 5x = 2\)**
- Rearranging gives:
\[
y = 5x + 2
\]

6. **\(y - 4x + 1 = 0\)**
- Rearranging gives:
\[
y = 4x - 1
\]

7. **\(y = 5x\)**
- This equation is already in slope-intercept form.

### Summary of the Equations in \(y = mx + b\) Format
1. \(y = 5x - 4\)
2. \(y = -2x + 3\)
3. \(y = 3x + 1\)
4. \(y = 3x + 1\) (same as equation 3)
5. \(y = 5x + 2\)
6. \(y = 4x - 1\)
7. \(y = 5x\)

### Step b: Identify Pairs of Equations for Systems

Now, we can proceed to identify which pairs of equations can form a system with one solution, no solution, or infinite solutions based on their slopes and intercepts.

1. **Equations 1 & 7**:
- \(y = 5x - 4\) and \(y = 5x\) (same slope, different y-intercepts) → No solution (parallel lines)

2. **Equations 2 & 3**:
- \(y = -2x + 3\) and \(y = 3x + 1\) (different slopes) → One solution (intersecting lines)

3. **Equations 4 & 3**:
- \(y = 3x + 1\) and \(y = 3x + 1\) (same slope and intercept) → Infinite solutions (same line)

4. **Equations 5 & 1**:
- \(y = 5x + 2\) and \(y = 5x - 4\) (same slope, different y-intercepts) → No solution (parallel lines)

5. **Equations 6 & 3**:
- \(y = 4x - 1\) and \(y = 3x + 1\) (different slopes) → One solution (intersecting lines)

### Step c: Graphing
When graphing these equations, the characteristics outlined will show:

- **One Solution**: Where two lines intersect.
- **No Solution**: Where two lines are parallel (never intersect).
- **Infinite Solutions**: Where two lines lie on top of each other (exactly the same).

### Step d: Conclusion
You can summarize the nature of systems formed by the given pairs:

- **1 & 7**: No solution
- **2 & 3**: One solution
- **4 & 3**: Infinite solutions
- **5 & 1**: No solution
- **6 & 3**: One solution

### Step e: Reflection
Reflect on the concepts of parallel lines, identical lines, and intersecting lines. These concepts illustrate how linear equations can graphically represent relationships between different variables.

### Please Proceed
If there are more questions or additional activities, feel free to ask!