Question

Given the following four lines pick the true statement

Line 1; 3y = 4x + 3
Line 2; for y = 3x - 4
Line 3; 3x + 4y = 8
Line 4; 4x + 3y = -6

A. Lines 1 and 4 are parallel
B. Lines two and three are parallel
C. Lines 2 and 4 perpendicular
D. Lines wanted to a perpendicular

Answer 1

Four questions five through eight, write a linear equation with the given information
**Formula is y-y1=m(x-x1)**

5. Passing through 2,3 in parallel to y = x - 4
6. passing through -6, -1 and parallel to 2x + 3y = 3
7. Passing through -2, 8 and perpendicular to y = 2x + 5
8. Passing through -2, -11 and perpendicular to x + 4y = 8

Answer 2

To determine which of the statements is true, let's first rewrite each line in the slope-intercept form \( y = mx + b \), where \( m \) is the slope.

1. **Line 1: \( 3y = 4x + 3 \)**
Divide by 3:
\[ y = \frac{4}{3}x + 1 \]
(Slope \( m_1 = \frac{4}{3} \))

2. **Line 2: \( y = 3x - 4 \)**
(Slope \( m_2 = 3 \))

3. **Line 3: \( 3x + 4y = 8 \)**
Rearranging gives:
\[ 4y = -3x + 8 \]
Divide by 4:
\[ y = -\frac{3}{4}x + 2 \]
(Slope \( m_3 = -\frac{3}{4} \))

4. **Line 4: \( 4x + 3y = -6 \)**
Rearranging gives:
\[ 3y = -4x - 6 \]
Divide by 3:
\[ y = -\frac{4}{3}x - 2 \]
(Slope \( m_4 = -\frac{4}{3} \))

Now, let's analyze the slopes to check for parallelism and perpendicularity:

- Lines are **parallel** if they have the same slope.
- **Line 1** \( \left(\frac{4}{3}\right) \) and **Line 4** \( \left(-\frac{4}{3}\right) \) have different slopes, so they are not parallel.
- **Line 2** \( (3) \) and **Line 3** \( \left(-\frac{3}{4}\right) \) have different slopes, so they are not parallel.

- Lines are **perpendicular** if the product of their slopes is -1.
- \( m_2 \left(3\right) \) and \( m_4 \left(-\frac{4}{3}\right) \):
\[ 3 \cdot -\frac{4}{3} = -4 \] (not -1, so not perpendicular)

- \( m_1 \left(\frac{4}{3}\right) \) and \( m_3 \left(-\frac{3}{4}\right) \):
\[ \frac{4}{3} \cdot -\frac{3}{4} = -1 \] (they are perpendicular)

So, the only correct statement is:

**D. Lines 1 and 3 are perpendicular.**

Answer 3

Let's tackle each question one by one, using the provided formula \( y - y_1 = m(x - x_1) \) for the equations.

### Question 5
**Passing through (2, 3) and parallel to \( y = x - 4 \)**

1. The slope (m) of the line \( y = x - 4 \) is 1 (since it can be rewritten as \( y = 1x - 4 \)).
2. Since the new line is parallel, it will also have a slope of 1.
3. Using the point-slope form:
\[
y - 3 = 1(x - 2)
\]
4. Simplifying:
\[
y - 3 = x - 2 \quad \Rightarrow \quad y = x + 1
\]

### Question 6
**Passing through (-6, -1) and parallel to \( 2x + 3y = 3 \)**

1. First, find the slope of the line \( 2x + 3y = 3 \). Rearranging into slope-intercept form (y = mx + b):
\[
3y = -2x + 3 \quad \Rightarrow \quad y = -\frac{2}{3}x + 1
\]
So, the slope (m) is \( -\frac{2}{3} \).

2. The new line will have the same slope, \( m = -\frac{2}{3} \).
3. Using the point-slope form:
\[
y - (-1) = -\frac{2}{3}(x - (-6))
\]
\[
y + 1 = -\frac{2}{3}(x + 6)
\]
4. Simplifying:
\[
y + 1 = -\frac{2}{3}x - 4 \quad \Rightarrow \quad y = -\frac{2}{3}x - 5
\]

### Question 7
**Passing through (-2, 8) and perpendicular to \( y = 2x + 5 \)**

1. The slope of the line \( y = 2x + 5 \) is 2.
2. The slope of the line perpendicular to it will be the negative reciprocal, so \( m = -\frac{1}{2} \).
3. Using the point-slope form:
\[
y - 8 = -\frac{1}{2}(x - (-2))
\]
\[
y - 8 = -\frac{1}{2}(x + 2)
\]
4. Simplifying:
\[
y - 8 = -\frac{1}{2}x - 1 \quad \Rightarrow \quad y = -\frac{1}{2}x + 7
\]

### Question 8
**Passing through (-2, -11) and perpendicular to \( x + 4y = 8 \)**

1. First, find the slope of the line \( x + 4y = 8 \). Rearranging gives:
\[
4y = -x + 8 \quad \Rightarrow \quad y = -\frac{1}{4}x + 2
\]
So, the slope (m) is \( -\frac{1}{4} \).

2. The slope of the line perpendicular to it is the negative reciprocal, which is \( m = 4 \).
3. Using the point-slope form:
\[
y - (-11) = 4(x - (-2))
\]
\[
y + 11 = 4(x + 2)
\]
4. Simplifying:
\[
y + 11 = 4x + 8 \quad \Rightarrow \quad y = 4x - 3
\]

### Summary of Linear Equations
5. \( y = x + 1 \)
6. \( y = -\frac{2}{3}x - 5 \)
7. \( y = -\frac{1}{2}x + 7 \)
8. \( y = 4x - 3 \)