Question

Graph the line given by -x + y = 1
*

Answer 1

Which of the following pairs of formulas will form the same line?
*
1 point
y = 2x + 4 and y - 6 = 2(x - 1)
y = 3x and 3x + 3y = 0
4x - 5y = 15 and y = (-4/5)x + 3
None of the Above
What is the slope of a horizontal line? Of a vertical line?
*
1 point
Constant Function: 3; Vertical Line: 2
Constant Function: 0; Vertical Line: Undefined
Constant Function: Undefined ; Vertical Line: Zero
None of the Above

Answer 2

Which of the following pairs of formulas will form the same line?
*
1 point
y = 2x + 4 and y - 6 = 2(x - 1)
y = 3x and 3x + 3y = 0
4x - 5y = 15 and y = (-4/5)x + 3
None of the Above

Answer 3

To graph the line given by the equation \(-x + y = 1\), we can follow these steps:

1. **Rearrange the equation (Optional)**: You can rewrite it in slope-intercept form \(y = mx + b\):
\[
y = x + 1
\]
This shows that the slope \(m\) is 1 and the y-intercept \(b\) is 1.

2. **Identify the y-intercept**: The y-intercept is the point where the line crosses the y-axis. From the equation, when \(x = 0\):
\[
y = 0 + 1 = 1
\]
So the point is \((0, 1)\).

3. **Identify another point using the slope**: Since the slope is 1, you can find another point. From the y-intercept \((0, 1)\), if you go up 1 unit (rise) and to the right 1 unit (run), you reach \((1, 2)\).

4. **Plot the points**: Plot the points \((0, 1)\) and \((1, 2)\) on a graph.

5. **Draw the line**: Draw a straight line through these points extending in both directions.

The resulting graph will represent the line described by the equation \(-x + y = 1\) or equivalently \(y = x + 1\).

If you're visualizing this, the line will pass through the points (0, 1) and (1, 2) and will have a positive slope.

Answer 4

Let's analyze each pair of formulas to see which forms the same line:

### Pair 1: \( y = 2x + 4 \) and \( y - 6 = 2(x - 1) \)

1. The first equation is already in slope-intercept form:
\[
y = 2x + 4
\]

2. Simplifying the second equation:
\[
y - 6 = 2(x - 1)
\]
\[
y - 6 = 2x - 2 \Rightarrow y = 2x + 4
\]
These two equations represent the same line.

---

### Pair 2: \( y = 3x \) and \( 3x + 3y = 0 \)

1. The first equation is already in slope-intercept form:
\[
y = 3x
\]

2. Now simplifying the second equation:
\[
3x + 3y = 0 \Rightarrow 3y = -3x \Rightarrow y = -x
\]
These two equations do **not** represent the same line.

---

### Pair 3: \( 4x - 5y = 15 \) and \( y = \left(-\frac{4}{5}\right)x + 3 \)

1. Rearranging the first equation:
\[
4x - 5y = 15 \Rightarrow -5y = -4x + 15 \Rightarrow y = \frac{4}{5}x - 3
\]

2. The second equation is:
\[
y = -\frac{4}{5}x + 3
\]
These two equations do **not** represent the same line.

---

### Conclusion for the line equations:
The only pair of formulas that represent the same line is:
**Pair 1: \( y = 2x + 4 \) and \( y - 6 = 2(x - 1) \)**.

---

### Slope of Horizontal and Vertical Lines

1. The slope of a **horizontal line** is always **0** because there is no change in the y-value as x changes (it runs parallel to the x-axis).

2. The slope of a **vertical line** is **undefined** because there is no change in the x-value as y changes (it runs parallel to the y-axis).

### Conclusion for the slope:
The correct statement is:
**Constant Function: 0; Vertical Line: Undefined**.

### Final answers:
1. **The first pair forms the same line.**
2. **Constant Function: 0; Vertical Line: Undefined**.

Answer 5

Let's go through each pair of formulas to determine if they represent the same line.

### Pair 1: \( y = 2x + 4 \) and \( y - 6 = 2(x - 1) \)

1. The first equation is already in slope-intercept form:
\[
y = 2x + 4
\]

2. For the second equation, let's simplify it:
\[
y - 6 = 2(x - 1)
\]
Distributing the 2:
\[
y - 6 = 2x - 2
\]
Adding 6 to both sides:
\[
y = 2x + 4
\]

**Conclusion**: Both equations represent the same line.

---

### Pair 2: \( y = 3x \) and \( 3x + 3y = 0 \)

1. The first equation is already in slope-intercept form:
\[
y = 3x
\]

2. Now simplify the second equation:
\[
3x + 3y = 0
\]
To isolate \(y\), we can subtract \(3x\) from both sides:
\[
3y = -3x
\]
Dividing both sides by 3:
\[
y = -x
\]

**Conclusion**: These two equations do **not** represent the same line.

---

### Pair 3: \( 4x - 5y = 15 \) and \( y = \left(-\frac{4}{5}\right)x + 3 \)

1. Rearranging the first equation:
\[
4x - 5y = 15
\]
To isolate \(y\):
\[
-5y = -4x + 15 \implies y = \frac{4}{5}x - 3
\]

2. The second equation is:
\[
y = -\frac{4}{5}x + 3
\]

**Conclusion**: These two equations do **not** represent the same line.

---

### Final Assessment

- Pair 1: **Represents the same line**.
- Pair 2: **Does not represent the same line**.
- Pair 3: **Does not represent the same line**.

**Answer**: The only pair of formulas that will form the same line is:
**\(y = 2x + 4\) and \(y - 6 = 2(x - 1)\)**.