Math / Geometry

QuestionComplete the sentence.
The lines $2 x+4 y=32$ and $y=-\frac{1}{2} x+16$ are perpendicular

Studdy Solution

STEP 1

What is this asking? Are these two lines perpendicular? Watch out! Don't forget, perpendicular lines have slopes that are negative reciprocals of each other!

STEP 2

1. Rewrite the first equation in slope-intercept form.
2. Compare the slopes.

STEP 3

We want to get the first equation to look like $y = mx + b$ , so let's isolate the $y$ term.
Starting with $2x + 4y = 32$ , we can subtract $2x$ from both sides.
This gives us $4y = -2x + 32$ .
Remember, we're subtracting $2x$ from both sides to move the $2x$ to the other side of the equation!

STEP 4

Now, we want to get $y$ all by itself.
Since $y$ is being multiplied by 4, we can divide both sides of the equation by 4!
This gives us $y = \frac{-2x + 32}{4}$ .
We're dividing everything on the right side by 4, so we get $y = -\frac{2}{4}x + \frac{32}{4}$ .

STEP 5

Let's simplify those fractions!
We have $-\frac{2}{4}$ , which simplifies to $-\frac{1}{2}$ , and $\frac{32}{4}$ , which simplifies to 8.
So, our equation becomes $y = -\frac{1}{2}x + 8$ .
Awesome!

STEP 6

The slope of the first line is now staring us in the face!
It's $-\frac{1}{2}$ .
The second equation is already in slope-intercept form, $y = -\frac{1}{2}x + 16$ , and its slope is also $-\frac{1}{2}$ .

STEP 7

Are the slopes negative reciprocals of each other?
For two slopes to be negative reciprocals, they need to multiply to -1.
Let's check: $\left(-\frac{1}{2}\right) \cdot \left(-\frac{1}{2}\right) = \frac{1}{4}$ .
Since $\frac{1}{4}$ is not equal to -1, these slopes are not negative reciprocals.

STEP 8

The lines are not perpendicular.
They are parallel!

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QuestionComplete the sentence.
The lines $2 x+4 y=32$ and $y=-\frac{1}{2} x+16$ are perpendicular

Studdy Solution

STEP 1

What is this asking? Are these two lines perpendicular? Watch out! Don't forget, perpendicular lines have slopes that are negative reciprocals of each other!

STEP 2

1. Rewrite the first equation in slope-intercept form.
2. Compare the slopes.

STEP 3

We want to get the first equation to look like $y = mx + b$ , so let's isolate the $y$ term.
Starting with $2x + 4y = 32$ , we can subtract $2x$ from both sides.
This gives us $4y = -2x + 32$ .
Remember, we're subtracting $2x$ from both sides to move the $2x$ to the other side of the equation!

STEP 4

Now, we want to get $y$ all by itself.
Since $y$ is being multiplied by 4, we can divide both sides of the equation by 4!
This gives us $y = \frac{-2x + 32}{4}$ .
We're dividing everything on the right side by 4, so we get $y = -\frac{2}{4}x + \frac{32}{4}$ .

STEP 5

Let's simplify those fractions!
We have $-\frac{2}{4}$ , which simplifies to $-\frac{1}{2}$ , and $\frac{32}{4}$ , which simplifies to 8.
So, our equation becomes $y = -\frac{1}{2}x + 8$ .
Awesome!

STEP 6

The slope of the first line is now staring us in the face!
It's $-\frac{1}{2}$ .
The second equation is already in slope-intercept form, $y = -\frac{1}{2}x + 16$ , and its slope is also $-\frac{1}{2}$ .

STEP 7

Are the slopes negative reciprocals of each other?
For two slopes to be negative reciprocals, they need to multiply to -1.
Let's check: $\left(-\frac{1}{2}\right) \cdot \left(-\frac{1}{2}\right) = \frac{1}{4}$ .
Since $\frac{1}{4}$ is not equal to -1, these slopes are not negative reciprocals.

STEP 8

The lines are not perpendicular.
They are parallel!

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QuestionComplete the sentence.The lines 2x+4y=322 x+4 y=322x+4y=32 and y=−12x+16y=-\frac{1}{2} x+16y=−21​x+16 are perpendicular

Studdy Solution

STEP 1

What is this asking? Are these two lines perpendicular? Watch out! Don't forget, perpendicular lines have slopes that are *negative reciprocals* of each other!

STEP 2

1. Rewrite the first equation in slope-intercept form.2. Compare the slopes.

STEP 3

STEP 4

STEP 5

Let's **simplify** those fractions!We have −24-\frac{2}{4}−42​, which simplifies to −12-\frac{1}{2}−21​, and 324\frac{32}{4}432​, which simplifies to **8**.So, our equation becomes y=−12x+8y = -\frac{1}{2}x + 8y=−21​x+8.Awesome!

STEP 6

The **slope** of the first line is now staring us in the face!It's −12-\frac{1}{2}−21​.The second equation is already in slope-intercept form, y=−12x+16y = -\frac{1}{2}x + 16y=−21​x+16, and its slope is *also* −12-\frac{1}{2}−21​.

STEP 7

STEP 8

The lines are *not* perpendicular.They are parallel!

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionComplete the sentence.
The lines $2 x+4 y=32$ and $y=-\frac{1}{2} x+16$ are perpendicular

What is this asking? Are these two lines perpendicular? Watch out! Don't forget, perpendicular lines have slopes that are negative reciprocals of each other!

1. Rewrite the first equation in slope-intercept form.
2. Compare the slopes.

Let's simplify those fractions!
We have $-\frac{2}{4}$ , which simplifies to $-\frac{1}{2}$ , and $\frac{32}{4}$ , which simplifies to 8.
So, our equation becomes $y = -\frac{1}{2}x + 8$ .
Awesome!

The slope of the first line is now staring us in the face!
It's $-\frac{1}{2}$ .
The second equation is already in slope-intercept form, $y = -\frac{1}{2}x + 16$ , and its slope is also $-\frac{1}{2}$ .

The lines are not perpendicular.
They are parallel!