Solved on Feb 06, 2024

If $y$ varies directly with $x$ , and $y=12$ when $x=6$ , find $y$ when $x=3$ .

STEP 1

Assumptions
1. The relationship between $y$ and $x$ is direct variation.
2. When $x = 6$ , $y = 12$ .
3. We need to find the value of $y$ when $x = 3$ .

STEP 2

Since $y$ varies directly as $x$ , we can write the relationship as $y = kx$ , where $k$ is the constant of variation.

STEP 3

We need to find the constant of variation $k$ . We can use the given values of $x$ and $y$ to do this.
$y = kx$

STEP 4

Substitute the given values $x = 6$ and $y = 12$ into the equation.
$12 = k \cdot 6$

STEP 5

Solve for $k$ by dividing both sides of the equation by 6.
$k = \frac{12}{6}$

STEP 6

Calculate the value of $k$ .
$k = 2$

STEP 7

Now that we have the constant of variation, we can use it to find $y$ when $x = 3$ .
$y = kx$

STEP 8

Substitute $k = 2$ and $x = 3$ into the equation.
$y = 2 \cdot 3$

STEP 9

Calculate the value of $y$ .
$y = 6$
The value of $y$ when $x = 3$ is $y = 6$ .

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If $y$ varies directly with $x$ , and $y=12$ when $x=6$ , find $y$ when $x=3$ .

STEP 1

Assumptions
1. The relationship between $y$ and $x$ is direct variation.
2. When $x = 6$ , $y = 12$ .
3. We need to find the value of $y$ when $x = 3$ .

STEP 2

Since $y$ varies directly as $x$ , we can write the relationship as $y = kx$ , where $k$ is the constant of variation.

STEP 3

We need to find the constant of variation $k$ . We can use the given values of $x$ and $y$ to do this.
$y = kx$

STEP 4

Substitute the given values $x = 6$ and $y = 12$ into the equation.
$12 = k \cdot 6$

STEP 5

Solve for $k$ by dividing both sides of the equation by 6.
$k = \frac{12}{6}$

STEP 6

Calculate the value of $k$ .
$k = 2$

STEP 7

Now that we have the constant of variation, we can use it to find $y$ when $x = 3$ .
$y = kx$

STEP 8

Substitute $k = 2$ and $x = 3$ into the equation.
$y = 2 \cdot 3$

STEP 9

Calculate the value of $y$ .
$y = 6$
The value of $y$ when $x = 3$ is $y = 6$ .

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If yyy varies directly with xxx, and y=12y=12y=12 when x=6x=6x=6, find yyy when x=3x=3x=3.

STEP 1

Assumptions1. The relationship between yyy and xxx is direct variation.2. When x=6x = 6x=6, y=12y = 12y=12.3. We need to find the value of yyy when x=3x = 3x=3.

STEP 2

Since yyy varies directly as xxx, we can write the relationship as y=kxy = kxy=kx, where kkk is the constant of variation.

STEP 3

We need to find the constant of variation kkk. We can use the given values of xxx and yyy to do this.y=kxy = kxy=kx

STEP 4

Substitute the given values x=6x = 6x=6 and y=12y = 12y=12 into the equation.12=k⋅612 = k \cdot 612=k⋅6

STEP 5

Solve for kkk by dividing both sides of the equation by 6.k=126k = \frac{12}{6}k=612​

STEP 6

Calculate the value of kkk.k=2k = 2k=2

STEP 7

Now that we have the constant of variation, we can use it to find yyy when x=3x = 3x=3.y=kxy = kxy=kx

STEP 8

Substitute k=2k = 2k=2 and x=3x = 3x=3 into the equation.y=2⋅3y = 2 \cdot 3y=2⋅3

STEP 9

Calculate the value of yyy.y=6y = 6y=6The value of yyy when x=3x = 3x=3 is y=6y = 6y=6.

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If $y$ varies directly with $x$ , and $y=12$ when $x=6$ , find $y$ when $x=3$ .

Assumptions
1. The relationship between $y$ and $x$ is direct variation.
2. When $x = 6$ , $y = 12$ .
3. We need to find the value of $y$ when $x = 3$ .

Since $y$ varies directly as $x$ , we can write the relationship as $y = kx$ , where $k$ is the constant of variation.

We need to find the constant of variation $k$ . We can use the given values of $x$ and $y$ to do this.
$y = kx$

Substitute the given values $x = 6$ and $y = 12$ into the equation.
$12 = k \cdot 6$

Solve for $k$ by dividing both sides of the equation by 6.
$k = \frac{12}{6}$

Calculate the value of $k$ .
$k = 2$

Now that we have the constant of variation, we can use it to find $y$ when $x = 3$ .
$y = kx$

Substitute $k = 2$ and $x = 3$ into the equation.
$y = 2 \cdot 3$

Calculate the value of $y$ .
$y = 6$
The value of $y$ when $x = 3$ is $y = 6$ .