Math / Algebra

Question27. $y=6$
28. $x=-3$
29. $y=-2$
30. $x=7$
Write each equation in standard form using integers.
31. $y=2 x+5$
32. $y+3=4(x-1)$ See Problem 4.
34. $y=\frac{1}{4} x-2$
35. $y=-\frac{2}{3} x-1$
33. $y-4=-2(x-3)$
36. $y+2=\frac{2}{3}(x+4)$
37. Video Games In a video game, you earn 5 points for each jewel you find. You earn 2 points for each star you find. Write and graph an equation that represents - See Problem 5. the numbers of jewels and stars you must find to earn 250 points. What are three combinations of jewels and stars you can find that will earn you 250 points?
38. Clothing A store sells T-shirts for $\$ 12$ each and sweatshirts for $\$ 15$ each. You

Studdy Solution

STEP 1

What is this asking? We need to rewrite the given equations in standard form using only integers.
Specifically, we're focusing on problems 34, 35, and 36. Watch out! Standard form means $Ax + By = C$ , where $A$ , $B$ , and $C$ are integers, and $A$ is non-negative!
Don't forget to multiply everything by the common denominator to get rid of those pesky fractions.

STEP 2

1. Rewrite equation 34 in standard form.
2. Rewrite equation 35 in standard form.
3. Rewrite equation 36 in standard form.

STEP 3

We're starting with $y = \frac{1}{4}x - 2$ .

STEP 4

To get rid of that $\frac{1}{4}$ , let's multiply both sides of the equation by 4.
This is like giving both sides an equal power-up!
This gives us $4 \cdot y = 4 \cdot (\frac{1}{4}x - 2)$ , which simplifies to $4y = x - 8$ .
Remember, $4 \cdot \frac{1}{4} = 1$ , and $4 \cdot -2 = -8$ .

STEP 5

We want it in the form $Ax + By = C$ .
So, we'll subtract $x$ from both sides to get $-x + 4y = -8$ .

STEP 6

Almost there!
We need $A$ to be positive, so we multiply both sides by $-1$ giving us $x - 4y = 8$ .
Boom! Standard form achieved.

STEP 7

Our starting equation is $y = -\frac{2}{3}x - 1$ .

STEP 8

Multiply both sides by 3 to banish the fraction: $3 \cdot y = 3 \cdot (-\frac{2}{3}x - 1)$ .
This simplifies to $3y = -2x - 3$ .

STEP 9

Let's add $2x$ to both sides to get $2x + 3y = -3$ .
Look at that, already in standard form!

STEP 10

We begin with $y + 2 = \frac{2}{3}(x + 4)$ .

STEP 11

Multiply both sides by 3: $3 \cdot (y + 2) = 3 \cdot \frac{2}{3}(x + 4)$ .
This simplifies to $3y + 6 = 2(x + 4)$ .

STEP 12

Distribute the 2 on the right side: $3y + 6 = 2x + 8$ .

STEP 13

Subtract $2x$ from both sides: $-2x + 3y + 6 = 8$ .
Then, subtract 6 from both sides: $-2x + 3y = 2$ .

STEP 14

Multiply both sides by $-1$ to make the coefficient of $x$ positive: $2x - 3y = -2$ .
And there we have it, standard form!

STEP 15

34. $x - 4y = 8$
35. $2x + 3y = -3$
36. $2x - 3y = -2$

Was this helpful?

Studdy Solution

STEP 1

STEP 2

1. Rewrite equation 34 in standard form.
2. Rewrite equation 35 in standard form.
3. Rewrite equation 36 in standard form.

STEP 3

We're starting with $y = \frac{1}{4}x - 2$ .

STEP 4

To get rid of that $\frac{1}{4}$ , let's multiply both sides of the equation by 4.
This is like giving both sides an equal power-up!
This gives us $4 \cdot y = 4 \cdot (\frac{1}{4}x - 2)$ , which simplifies to $4y = x - 8$ .
Remember, $4 \cdot \frac{1}{4} = 1$ , and $4 \cdot -2 = -8$ .

STEP 5

We want it in the form $Ax + By = C$ .
So, we'll subtract $x$ from both sides to get $-x + 4y = -8$ .

STEP 6

Almost there!
We need $A$ to be positive, so we multiply both sides by $-1$ giving us $x - 4y = 8$ .
Boom! Standard form achieved.

STEP 7

Our starting equation is $y = -\frac{2}{3}x - 1$ .

STEP 8

Multiply both sides by 3 to banish the fraction: $3 \cdot y = 3 \cdot (-\frac{2}{3}x - 1)$ .
This simplifies to $3y = -2x - 3$ .

STEP 9

Let's add $2x$ to both sides to get $2x + 3y = -3$ .
Look at that, already in standard form!

STEP 10

We begin with $y + 2 = \frac{2}{3}(x + 4)$ .

STEP 11

Multiply both sides by 3: $3 \cdot (y + 2) = 3 \cdot \frac{2}{3}(x + 4)$ .
This simplifies to $3y + 6 = 2(x + 4)$ .

STEP 12

Distribute the 2 on the right side: $3y + 6 = 2x + 8$ .

STEP 13

Subtract $2x$ from both sides: $-2x + 3y + 6 = 8$ .
Then, subtract 6 from both sides: $-2x + 3y = 2$ .

STEP 14

Multiply both sides by $-1$ to make the coefficient of $x$ positive: $2x - 3y = -2$ .
And there we have it, standard form!

STEP 15

34. $x - 4y = 8$
35. $2x + 3y = -3$
36. $2x - 3y = -2$

Get Studdy

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.

Studdy Solution

STEP 1

STEP 2

1. Rewrite equation 34 in standard form.2. Rewrite equation 35 in standard form.3. Rewrite equation 36 in standard form.

STEP 3

We're starting with y=14x−2y = \frac{1}{4}x - 2y=41​x−2.

STEP 4

STEP 5

We want it in the form Ax+By=CAx + By = CAx+By=C.So, we'll subtract xxx from both sides to get −x+4y=−8-x + 4y = -8−x+4y=−8.

STEP 6

Almost there!We need AAA to be positive, so we multiply *both* sides by −1-1−1 giving us x−4y=8x - 4y = 8x−4y=8.Boom! Standard form achieved.

STEP 7

Our starting equation is y=−23x−1y = -\frac{2}{3}x - 1y=−32​x−1.

STEP 8

Multiply *both* sides by **3** to banish the fraction: 3⋅y=3⋅(−23x−1)3 \cdot y = 3 \cdot (-\frac{2}{3}x - 1)3⋅y=3⋅(−32​x−1).This simplifies to 3y=−2x−33y = -2x - 33y=−2x−3.

STEP 9

Let's add 2x2x2x to both sides to get 2x+3y=−32x + 3y = -32x+3y=−3.Look at that, already in standard form!

STEP 10

We begin with y+2=23(x+4)y + 2 = \frac{2}{3}(x + 4)y+2=32​(x+4).

STEP 11

Multiply *both* sides by **3**: 3⋅(y+2)=3⋅23(x+4)3 \cdot (y + 2) = 3 \cdot \frac{2}{3}(x + 4)3⋅(y+2)=3⋅32​(x+4).This simplifies to 3y+6=2(x+4)3y + 6 = 2(x + 4)3y+6=2(x+4).

STEP 12

Distribute the **2** on the right side: 3y+6=2x+83y + 6 = 2x + 83y+6=2x+8.

STEP 13

Subtract 2x2x2x from both sides: −2x+3y+6=8-2x + 3y + 6 = 8−2x+3y+6=8.Then, subtract **6** from both sides: −2x+3y=2-2x + 3y = 2−2x+3y=2.

STEP 14

Multiply both sides by −1-1−1 to make the coefficient of xxx positive: 2x−3y=−22x - 3y = -22x−3y=−2.And there we have it, standard form!

STEP 15

34. x−4y=8x - 4y = 8x−4y=835. 2x+3y=−32x + 3y = -32x+3y=−336. 2x−3y=−22x - 3y = -22x−3y=−2

Get Studdy

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.

1. Rewrite equation 34 in standard form.
2. Rewrite equation 35 in standard form.
3. Rewrite equation 36 in standard form.

We're starting with $y = \frac{1}{4}x - 2$ .

We want it in the form $Ax + By = C$ .
So, we'll subtract $x$ from both sides to get $-x + 4y = -8$ .

Almost there!
We need $A$ to be positive, so we multiply both sides by $-1$ giving us $x - 4y = 8$ .
Boom! Standard form achieved.

Our starting equation is $y = -\frac{2}{3}x - 1$ .

Multiply both sides by 3 to banish the fraction: $3 \cdot y = 3 \cdot (-\frac{2}{3}x - 1)$ .
This simplifies to $3y = -2x - 3$ .

Let's add $2x$ to both sides to get $2x + 3y = -3$ .
Look at that, already in standard form!

We begin with $y + 2 = \frac{2}{3}(x + 4)$ .

Multiply both sides by 3: $3 \cdot (y + 2) = 3 \cdot \frac{2}{3}(x + 4)$ .
This simplifies to $3y + 6 = 2(x + 4)$ .

Distribute the 2 on the right side: $3y + 6 = 2x + 8$ .

Subtract $2x$ from both sides: $-2x + 3y + 6 = 8$ .
Then, subtract 6 from both sides: $-2x + 3y = 2$ .

Multiply both sides by $-1$ to make the coefficient of $x$ positive: $2x - 3y = -2$ .
And there we have it, standard form!

34. $x - 4y = 8$
35. $2x + 3y = -3$
36. $2x - 3y = -2$