### Turn 1 Question
equations of parallel & perpendicular lines mix-up!
$y = 2x$
$y = \frac{1}{4}x + 4$
$y = 5x$
$y = -\frac{1}{2}x + 3$
$y = -\frac{1}{2}x + 4$
$y = -\frac{1}{2}x + 1$
$y = -\frac{1}{2}x + 2$
$y = 2x - 1$
$y = -4x + 3$
$y = 5x - 20$
start
you made it!
through: $(1, 5)$, parallel to $y = 5x - 4$
through: $(-2, 4)$, parallel to $y = -\frac{1}{2}x + 4$
through: $(-2, 5)$, perp. to $y = 2x + 1$
through: $(-1, -2)$, perp. to $y = -\frac{1}{2}x + 1$
through: $(4, 5)$, perp. to $y = -4x + 2$
through: $(4, -1)$, parallel to $y = -\frac{1}{2}x + 2$
through: $(2, -5)$, parallel to $y = -4x + 1$
through: $(-2, 3)$, perp. to $y = 2x - 5$
through: $(-1, -3)$, perp. to $y = -\frac{1}{2}x + 2$
through: $(3, -5)$, parallel to $y = 5x - 3$

### Turn 2 Question
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Question

### Turn 1 Question
equations of parallel & perpendicular lines mix-up!
$y = 2x$
$y = \frac{1}{4}x + 4$
$y = 5x$
$y = -\frac{1}{2}x + 3$
$y = -\frac{1}{2}x + 4$
$y = -\frac{1}{2}x + 1$
$y = -\frac{1}{2}x + 2$
$y = 2x - 1$
$y = -4x + 3$
$y = 5x - 20$
start
you made it!
through: $(1, 5)$, parallel to $y = 5x - 4$
through: $(-2, 4)$, parallel to $y = -\frac{1}{2}x + 4$
through: $(-2, 5)$, perp. to $y = 2x + 1$
through: $(-1, -2)$, perp. to $y = -\frac{1}{2}x + 1$
through: $(4, 5)$, perp. to $y = -4x + 2$
through: $(4, -1)$, parallel to $y = -\frac{1}{2}x + 2$
through: $(2, -5)$, parallel to $y = -4x + 1$
through: $(-2, 3)$, perp. to $y = 2x - 5$
through: $(-1, -3)$, perp. to $y = -\frac{1}{2}x + 2$
through: $(3, -5)$, parallel to $y = 5x - 3$

### Turn 2 Question
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Accepted Answer

### Turn 1 Answer
Let's solve the problem "through: $(3, -5)$, parallel to $y = 5x - 3$" as an example (the first "START" problem).

# Explanation:
## Step1: Recall the slope of parallel lines
Parallel lines have the same slope. The given line is $y = 5x - 3$, which is in slope - intercept form $y=mx + b$ (where $m$ is the slope and $b$ is the y - intercept). So the slope $m$ of the line we want to find is also $m = 5$.

## Step2: Use the point - slope form of a line
The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)$ is a point on the line and $m$ is the slope. We know that $x_1 = 3$, $y_1=-5$ and $m = 5$. Substitute these values into the point - slope form:
$y-(-5)=5(x - 3)$

## Step3: Simplify the equation
Simplify the left - hand side: $y + 5=5(x - 3)$
Expand the right - hand side: $y+5 = 5x-15$
Subtract 5 from both sides to get the slope - intercept form: $y=5x-15 - 5$
$y = 5x-20$

Let's solve another problem, for example, "through: $(1,5)$, parallel to $y = 5x-4$":

## Step1: Determine the slope
The slope of the line $y = 5x - 4$ is $m = 5$ (since it is in $y=mx + b$ form). For a line parallel to this line, the slope of the new line is also $m = 5$.

## Step2: Use point - slope form
Using the point $(x_1,y_1)=(1,5)$ and $m = 5$ in the point - slope form $y - y_1=m(x - x_1)$:
$y - 5=5(x - 1)$

## Step3: Simplify the equation
Expand the right - hand side: $y - 5=5x-5$
Add 5 to both sides: $y=5x-5 + 5$
$y = 5x$

For the problem "through: $(-2,4)$, parallel to $y=-\frac{1}{2}x + 4$":

## Step1: Find the slope
The slope of the line $y =-\frac{1}{2}x + 4$ is $m=-\frac{1}{2}$. A line parallel to it has the same slope $m =-\frac{1}{2}$.

## Step2: Apply point - slope form
Using the point $(x_1,y_1)=(-2,4)$ and $m =-\frac{1}{2}$ in $y - y_1=m(x - x_1)$:
$y - 4=-\frac{1}{2}(x-(-2))$
$y - 4=-\frac{1}{2}(x + 2)$

## Step3: Simplify the equation
Expand the right - hand side: $y - 4=-\frac{1}{2}x-1$
Add 4 to both sides: $y=-\frac{1}{2}x-1 + 4$
$y=-\frac{1}{2}x + 3$

For the problem "through: $(-2,5)$, perp. to $y = 2x+1$":

## Step1: Find the slope of the perpendicular line
If two lines are perpendicular, the product of their slopes is $- 1$. Let the slope of the given line $y = 2x + 1$ be $m_1=2$, and the slope of the line we want to find be $m_2$. Then $m_1\times m_2=-1$, so $2\times m_2=-1$, and $m_2=-\frac{1}{2}$.

## Step2: Use point - slope form
Using the point $(x_1,y_1)=(-2,5)$ and $m =-\frac{1}{2}$ in $y - y_1=m(x - x_1)$:
$y - 5=-\frac{1}{2}(x-(-2))$
$y - 5=-\frac{1}{2}(x + 2)$

## Step3: Simplify the equation
Expand the right - hand side: $y - 5=-\frac{1}{2}x-1$
Add 5 to both sides: $y=-\frac{1}{2}x-1 + 5$
$y=-\frac{1}{2}x + 4$

For the problem "through: $(-1,-2)$, perp. to $y=-\frac{1}{2}x + 1$":

## Step1: Calculate the slope of the perpendicular line
The slope of the line $y=-\frac{1}{2}x + 1$ is $m_1 =-\frac{1}{2}$. Let the slope of the perpendicular line be $m_2$. Since $m_1\times m_2=-1$, we have $-\frac{1}{2}\times m_2=-1$, so $m_2 = 2$.

## Step2: Use point - slope form
Using the point $(x_1,y_1)=(-1,-2)$ and $m = 2$ in $y - y_1=m(x - x_1)$:
$y-(-2)=2(x-(-1))$
$y + 2=2(x + 1)$

## Step3: Simplify the equation
Expand the right - hand side: $y + 2=2x+2$
Subtract 2 from both sides: $y=2x+2 - 2$
$y = 2x$

For the problem "through: $(4,5)$, perp. to $y=-4x + 2$":

## Step1: Find the slope of the perpendicular line
The slope of the line $y=-4x + 2$ is $m_1=-4$. Let the slope of the perpendicular line be $m_2$. Since $m_1\times m_2=-1$, we have $-4\times m_2=-1$, so $m_2=\frac{1}{4}$.

## Step2: Use point - slope form
Using the point $(x_1,y_1)=(4,5)$ and $m=\frac{1}{4}$ in $y - y_1=m(x - x_1)$:
$y - 5=\frac{1}{4}(x - 4)$

## Step3: Simplify the equation
Expand the right - hand side: $y - 5=\frac{1}{4}x-1$
Add 5 to both sides: $y=\frac{1}{4}x-1 + 5$
$y=\frac{1}{4}x + 4$

For the problem "through: $(4,-1)$, parallel to $y=-\frac{1}{2}x + 2$":

## Step1: Determine the slope
The slope of the line $y=-\frac{1}{2}x + 2$ is $m =-\frac{1}{2}$. A line parallel to it has the same slope $m=-\frac{1}{2}$.

## Step2: Use point - slope form
Using the point $(x_1,y_1)=(4,-1)$ and $m =-\frac{1}{2}$ in $y - y_1=m(x - x_1)$:
$y-(-1)=-\frac{1}{2}(x - 4)$
$y + 1=-\frac{1}{2}(x - 4)$

## Step3: Simplify the equation
Expand the right - hand side: $y + 1=-\frac{1}{2}x+2$
Subtract 1 from both sides: $y=-\frac{1}{2}x+2 - 1$
$y=-\frac{1}{2}x + 1$

For the problem "through: $(2,-5)$, parallel to $y=-4x + 1$":

## Step1: Find the slope
The slope of the line $y=-4x + 1$ is $m=-4$. A line parallel to it has the same slope $m=-4$.

## Step2: Use point - slope form
Using the point $(x_1,y_1)=(2,-5)$ and $m=-4$ in $y - y_1=m(x - x_1)$:
$y-(-5)=-4(x - 2)$
$y + 5=-4(x - 2)$

## Step3: Simplify the equation
Expand the right - hand side: $y + 5=-4x + 8$
Subtract 5 from both sides: $y=-4x+8 - 5$
$y=-4x + 3$

For the problem "through: $(-2,3)$, perp. to $y = 2x-5$":

## Step1: Calculate the slope of the perpendicular line
The slope of the line $y = 2x-5$ is $m_1 = 2$. Let the slope of the perpendicular line be $m_2$. Since $m_1\times m_2=-1$, we have $2\times m_2=-1$, so $m_2=-\frac{1}{2}$.

## Step2: Use point - slope form
Using the point $(x_1,y_1)=(-2,3)$ and $m =-\frac{1}{2}$ in $y - y_1=m(x - x_1)$:
$y - 3=-\frac{1}{2}(x-(-2))$
$y - 3=-\frac{1}{2}(x + 2)$

## Step3: Simplify the equation
Expand the right - hand side: $y - 3=-\frac{1}{2}x-1$
Add 3 to both sides: $y=-\frac{1}{2}x-1 + 3$
$y=-\frac{1}{2}x + 2$

For the problem "through: $(-1,-3)$, perp. to $y=-\frac{1}{2}x + 2$":

## Step1: Find the slope of the perpendicular line
The slope of the line $y=-\frac{1}{2}x + 2$ is $m_1=-\frac{1}{2}$. Let the slope of the perpendicular line be $m_2$. Since $m_1\times m_2=-1$, we have $-\frac{1}{2}\times m_2=-1$, so $m_2 = 2$.

## Step2: Use point - slope form
Using the point $(x_1,y_1)=(-1,-3)$ and $m = 2$ in $y - y_1=m(x - x_1)$:
$y-(-3)=2(x-(-1))$
$y + 3=2(x + 1)$

## Step3: Simplify the equation
Expand the right - hand side: $y + 3=2x+2$
Subtract 3 from both sides: $y=2x+2 - 3$
$y=2x-1$

And for the "You MADE IT!" problem (let's assume we work backwards from the last one we solved to the start):

We can see that when we solve all these problems, we are matching the left - hand side equations with the right - hand side problem descriptions. For example, the solution to "through: $(3,-5)$, parallel to $y = 5x-3$" is $y = 5x-20$, the solution to "through: $(1,5)$, parallel to $y = 5x-4$" is $y = 5x$, the solution to "through: $(-2,4)$, parallel to $y=-\frac{1}{2}x + 4$" is $y=-\frac{1}{2}x + 3$, the solution to "through: $(-2,5)$, perp. to $y = 2x+1$" is $y=-\frac{1}{2}x + 4$, the solution to "through: $(-1,-2)$, perp. to $y=-\frac{1}{2}x + 1$" is $y = 2x$, the solution to "through: $(4,5)$, perp. to $y=-4x + 2$" is $y=\frac{1}{4}x + 4$, the solution to "through: $(4,-1)$, parallel to $y=-\frac{1}{2}x + 2$" is $y=-\frac{1}{2}x + 1$, the solution to "through: $(2,-5)$, parallel to $y=-4x + 1$" is $y=-4x + 3$, the solution to "through: $(-2,3)$, perp. to $y = 2x-5$" is $y=-\frac{1}{2}x + 2$, the solution to "through: $(-1,-3)$, perp. to $y=-\frac{1}{2}x + 2$" is $y=2x-1$

If we follow the path of matching the equations on the left with the problem descriptions on the right, we can see that the first problem (START) $y = 5x-20$ is the solution for "through: $(3,-5)$, parallel to $y = 5x-3$", and then we can follow the chain of parallel and perpendicular line equations to solve all the problems.

# Answers (for each sub - problem):
1. Through $(3,-5)$, parallel to $y = 5x-3$: $y = 5x-20$
2. Through $(1,5)$, parallel to $y = 5x-4$: $y = 5x$
3. Through $(-2,4)$, parallel to $y=-\frac{1}{2}x + 4$: $y=-\frac{1}{2}x + 3$
4. Through $(-2,5)$, perp. to $y = 2x+1$: $y=-\frac{1}{2}x + 4$
5. Through $(-1,-2)$, perp. to $y=-\frac{1}{2}x + 1$: $y = 2x$
6. Through $(4,5)$, perp. to $y=-4x + 2$: $y=\frac{1}{4}x + 4$
7. Through $(4,-1)$, parallel to $y=-\frac{1}{2}x + 2$: $y=-\frac{1}{2}x + 1$
8. Through $(2,-5)$, parallel to $y=-4x + 1$: $y=-4x + 3$
9. Through $(-2,3)$, perp. to $y = 2x-5$: $y=-\frac{1}{2}x + 2$
10. Through $(-1,-3)$, perp. to $y=-\frac{1}{2}x + 2$: $y=2x-1$

### Turn 2 Answer
Let me double-check the matches for you! Let's go one by one to confirm:

1. **through: (3, -5), parallel to y = 5x - 3** → Parallel lines have the same slope (5). Plugging into point-slope: y + 5 = 5(x - 3) → y = 5x - 20 ✔️ (matches the left column)
2. **through: (1, 5), parallel to y = 5x - 4** → Slope = 5. Point-slope: y - 5 = 5(x - 1) → y = 5x ✔️
3. **through: (-2, 4), parallel to y = -1/2x + 4** → Slope = -1/2. Point-slope: y - 4 = -1/2(x + 2) → y = -1/2x + 3 ✔️
4. **through: (-2, 5), perp. to y = 2x + 1** → Perpendicular slope is -1/2. Point-slope: y - 5 = -1/2(x + 2) → y = -1/2x + 4 ✔️
5. **through: (-1, -2), perp. to y = -1/2x + 1** → Perpendicular slope is 2. Point-slope: y + 2 = 2(x + 1) → y = 2x ✔️
6. **through: (4, 5), perp. to y = -4x + 2** → Perpendicular slope is 1/4. Point-slope: y - 5 = 1/4(x - 4) → y = 1/4x + 4 ✔️
7. **through: (4, -1), parallel to y = -1/2x + 2** → Slope = -1/2. Point-slope: y + 1 = -1/2(x - 4) → y = -1/2x + 1 ✔️
8. **through: (2, -5), parallel to y = -4x + 1** → Slope = -4. Point-slope: y + 5 = -4(x - 2) → y = -4x + 3 ✔️
9. **through: (-2, 3), perp. to y = 2x - 5** → Perpendicular slope is -1/2. Point-slope: y - 3 = -1/2(x + 2) → y = -1/2x + 2 ✔️
10. **through: (-1, -3), perp. to y = -1/2x + 2** → Perpendicular slope is 2. Point-slope: y + 3 = 2(x + 1) → y = 2x - 1 ✔️

All the matches are perfect! You nailed every single one. Nice work 😎

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Turn 1 Answer

Step1: Recall the slope of parallel lines

Step2: Use the point - slope form of a line

Step3: Simplify the equation

Step1: Determine the slope

Step2: Use point - slope form

Step3: Simplify the equation

Step1: Find the slope

Step2: Apply point - slope form

Step3: Simplify the equation

Step1: Find the slope of the perpendicular line

Step2: Use point - slope form

Step3: Simplify the equation

Step1: Calculate the slope of the perpendicular line

Step2: Use point - slope form

Step3: Simplify the equation

Step1: Find the slope of the perpendicular line

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答え

Response

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Explanation

Step1: Recall the slope of parallel lines

Step2: Use the point - slope form of a line

Step3: Simplify the equation

Step1: Determine the slope

Step2: Use point - slope form

Step3: Simplify the equation

Step1: Find the slope

Step2: Apply point - slope form

Step3: Simplify the equation

Step1: Find the slope of the perpendicular line

Step2: Use point - slope form

Step3: Simplify the equation

Step1: Calculate the slope of the perpendicular line

Step2: Use point - slope form

Step3: Simplify the equation

Step1: Find the slope of the perpendicular line

Step2: Use point - slope form

Step3: Simplify the equation

Step1: Determine the slope

Step2: Use point - slope form

Step3: Simplify the equation

Step1: Find the slope

Step2: Use point - slope form

Step3: Simplify the equation

Step1: Calculate the slope of the perpendicular line

Step2: Use point - slope form

Step3: Simplify the equation

Step1: Find the slope of the perpendicular line

Step2: Use point - slope form

Step3: Simplify the equation

Answer

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