tell whether each relationship is a variation.
8.
| x | -1 | 2 | 5 | 9 |
| y | 4 | 7 | 10 | 14 |
9.
| x | -2 | 2 | 6 | 10 |
| y | 1 | -1 | -3 | -5 |
10. write the equation $2x - 2y = 4$ in slope-intercept form, and then graph.
11. graph the line with slope $\frac{1}{3}$ that contains the point $(-4, -3)$.
12. write an equation in slope-intercept form for the line through $(-1, 1)$ and $(0, 3)$.
13. identify which lines are parallel: $y = -\frac{1}{2}x + 3$; $y = \frac{1}{2}x + 1$; $y = 2x$; $x + 2y = 4$.
14. identify which lines are perpendicular: $y - 2 = 3x$; $y + 4x = -1$; $y = -\frac{1}{3}x + 5$; $y = \frac{1}{3}x - 4$.
15. write an equation in slope-intercept form for the line that passes through $(0, 6)$ and is parallel to the line described by $y = 2x + 3$.
16. write an equation in slope-intercept form for the line that passes through $(4, 6)$ and is perpendicular to the line described by $y = x - 3$.

Question

Accepted Answer

### Question 10:
# Explanation:
## Step1: Recall slope - intercept form
The slope - intercept form of a linear equation is $y=mx + b$, where $m$ is the slope and $b$ is the y - intercept. We need to solve the equation $2x-2y = 4$ for $y$.
## Step2: Isolate the $y$ term
Subtract $2x$ from both sides of the equation: $- 2y=-2x + 4$.
## Step3: Solve for $y$
Divide each term in the equation $-2y=-2x + 4$ by $-2$. We get $y=\frac{-2x}{-2}+\frac{4}{-2}$, which simplifies to $y = x-2$.
# Answer:
The equation in slope - intercept form is $y=x - 2$. To graph it, the y - intercept $b=-2$, so the line crosses the y - axis at $(0,-2)$. The slope $m = 1$, so from the point $(0,-2)$, we can move 1 unit up and 1 unit to the right (or 1 unit down and 1 unit to the left) to find other points on the line.

### Question 12:
# Explanation:
## Step1: Find the slope
The slope $m$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by the formula $m=\frac{y_2 - y_1}{x_2 - x_1}$. For the points $(-1,1)$ and $(0,3)$, $x_1=-1,y_1 = 1,x_2 = 0,y_2=3$. Then $m=\frac{3 - 1}{0-(-1)}=\frac{2}{1}=2$.
## Step2: Find the y - intercept
The slope - intercept form is $y=mx + b$. We know that the line passes through the point $(0,3)$. When $x = 0$, $y=b$. So $b = 3$.
## Step3: Write the equation
Substitute $m = 2$ and $b = 3$ into the slope - intercept form $y=mx + b$. We get $y=2x+3$.
# Answer:
The equation in slope - intercept form is $y = 2x+3$.

### Question 13:
# Explanation:
## Step1: Recall the condition for parallel lines
Two non - vertical lines are parallel if and only if their slopes are equal. First, we need to write all the equations in slope - intercept form ($y=mx + b$) to identify their slopes.
- The equation $y=-\frac{1}{2}x + 3$ is in slope - intercept form with slope $m_1=-\frac{1}{2}$.
- The equation $y=\frac{1}{2}x + 1$ is in slope - intercept form with slope $m_2=\frac{1}{2}$.
- The equation $y = 2x$ is in slope - intercept form with slope $m_3=2$.
- For the equation $x + 2y=4$, solve for $y$: $2y=-x + 4$, then $y=-\frac{1}{2}x+2$. Its slope $m_4 =-\frac{1}{2}$.

## Step2: Identify parallel lines
Since the slope of $y =-\frac{1}{2}x + 3$ ($m_1=-\frac{1}{2}$) is equal to the slope of $x + 2y=4$ ($m_4=-\frac{1}{2}$), the lines $y=-\frac{1}{2}x + 3$ and $x + 2y=4$ are parallel.
# Answer:
The lines $y =-\frac{1}{2}x+3$ and $x + 2y = 4$ (or $y=-\frac{1}{2}x + 2$) are parallel.

### Question 14:
# Explanation:
## Step1: Recall the condition for perpendicular lines
Two non - vertical lines are perpendicular if the product of their slopes is $- 1$ (i.e., $m_1\times m_2=-1$). First, write all equations in slope - intercept form.
- For the equation $y - 2=3x$, we can rewrite it as $y=3x + 2$, so the slope $m_1 = 3$.
- For the equation $y+4x=-1$, we rewrite it as $y=-4x - 1$, so the slope $m_2=-4$.
- The equation $y=-\frac{1}{3}x + 5$ has slope $m_3=-\frac{1}{3}$.
- The equation $y=\frac{1}{3}x-4$ has slope $m_4=\frac{1}{3}$.

## Step2: Check the product of slopes
- Check $m_1$ and $m_3$: $m_1\times m_3=3\times(-\frac{1}{3})=-1$.

So the line $y - 2 = 3x$ (or $y=3x + 2$) and the line $y=-\frac{1}{3}x + 5$ are perpendicular.
# Answer:
The lines $y-2 = 3x$ (or $y = 3x+2$) and $y=-\frac{1}{3}x + 5$ are perpendicular.

### Question 15:
# Explanation:
## Step1: Recall the slope of parallel lines
If two lines are parallel, they have the same slope. The line $y = 2x+3$ has a slope $m = 2$. So the line we want to find also has a slope $m = 2$.
## Step2: Find the y - intercept
The slope - intercept form is $y=mx + b$. The line passes through the point $(0,6)$. When $x = 0$, $y=b$. So $b = 6$.
## Step3: Write the equation
Substitute $m = 2$ and $b = 6$ into $y=mx + b$. We get $y=2x+6$.
# Answer:
The equation in slope - intercept form is $y=2x + 6$.

### Question 16:
# Explanation:
## Step1: Recall the slope of perpendicular lines
If two lines are perpendicular, the product of their slopes is $-1$. The line $y=x - 3$ has a slope $m_1 = 1$. Let the slope of the perpendicular line be $m_2$. Then $m_1\times m_2=-1$, so $1\times m_2=-1$, which gives $m_2=-1$.
## Step2: Use the point - slope form to find the equation
The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(4,6)$ and $m=-1$. Substitute these values into the formula: $y - 6=-1(x - 4)$.
## Step3: Convert to slope - intercept form
Simplify $y - 6=-x + 4$. Add 6 to both sides: $y=-x+4 + 6$, so $y=-x + 10$.
# Answer:
The equation in slope - intercept form is $y=-x + 10$.

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QUESTION IMAGE

Question

Explanation:

Question 10:

Step1: Recall slope - intercept form

Step2: Isolate the \(y\) term

Step3: Solve for \(y\)

Step1: Find the slope

Step2: Find the y - intercept

Step3: Write the equation

Step1: Recall the condition for parallel lines

Step2: Identify parallel lines

Answer:

Question 12: