Solved on Jan 09, 2024

Find the direct variation equation passing through the point $(4,2)$ . Determine the constant of proportionality $\mathrm{k}$ .

STEP 1

Assumptions
1. The relationship between the variables $x$ and $y$ is a direct variation.
2. A direct variation can be described by the equation $y = kx$ , where $k$ is the constant of variation.
3. The given point that lies on the direct variation graph is $(4, 2)$ .

STEP 2

To find the constant of variation $k$ , we use the equation of direct variation and the given point.
$y = kx$

STEP 3

Substitute the coordinates of the given point $(4, 2)$ into the direct variation equation.
$2 = k \cdot 4$

STEP 4

Solve for the constant of variation $k$ by dividing both sides of the equation by 4.
$k = \frac{2}{4}$

STEP 5

Simplify the fraction to find the value of $k$ .
$k = \frac{1}{2}$

STEP 6

Now that we have the value of $k$ , we can write the equation of the direct variation.
$y = \frac{1}{2}x$
The equation of the direct variation that includes the given point $(4, 2)$ is $y = \frac{1}{2}x$ , and the constant of variation $\mathrm{k}$ is $\frac{1}{2}$ .

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Find the direct variation equation passing through the point $(4,2)$ . Determine the constant of proportionality $\mathrm{k}$ .

STEP 1

Assumptions
1. The relationship between the variables $x$ and $y$ is a direct variation.
2. A direct variation can be described by the equation $y = kx$ , where $k$ is the constant of variation.
3. The given point that lies on the direct variation graph is $(4, 2)$ .

STEP 2

To find the constant of variation $k$ , we use the equation of direct variation and the given point.
$y = kx$

STEP 3

Substitute the coordinates of the given point $(4, 2)$ into the direct variation equation.
$2 = k \cdot 4$

STEP 4

Solve for the constant of variation $k$ by dividing both sides of the equation by 4.
$k = \frac{2}{4}$

STEP 5

Simplify the fraction to find the value of $k$ .
$k = \frac{1}{2}$

STEP 6

Now that we have the value of $k$ , we can write the equation of the direct variation.
$y = \frac{1}{2}x$
The equation of the direct variation that includes the given point $(4, 2)$ is $y = \frac{1}{2}x$ , and the constant of variation $\mathrm{k}$ is $\frac{1}{2}$ .

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Find the direct variation equation passing through the point (4,2)(4,2)(4,2). Determine the constant of proportionality k\mathrm{k}k.

STEP 1

Assumptions1. The relationship between the variables xxx and yyy is a direct variation.2. A direct variation can be described by the equation y=kxy = kxy=kx, where kkk is the constant of variation.3. The given point that lies on the direct variation graph is (4,2)(4, 2)(4,2).

STEP 2

To find the constant of variation kkk, we use the equation of direct variation and the given point.y=kxy = kxy=kx

STEP 3

Substitute the coordinates of the given point (4,2)(4, 2)(4,2) into the direct variation equation.2=k⋅42 = k \cdot 42=k⋅4

STEP 4

Solve for the constant of variation kkk by dividing both sides of the equation by 4.k=24k = \frac{2}{4}k=42​

STEP 5

Simplify the fraction to find the value of kkk.k=12k = \frac{1}{2}k=21​

STEP 6

Now that we have the value of kkk, we can write the equation of the direct variation.y=12xy = \frac{1}{2}xy=21​xThe equation of the direct variation that includes the given point (4,2)(4, 2)(4,2) is y=12xy = \frac{1}{2}xy=21​x, and the constant of variation k\mathrm{k}k is 12\frac{1}{2}21​.

Start learning now

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

Find the direct variation equation passing through the point $(4,2)$ . Determine the constant of proportionality $\mathrm{k}$ .

Assumptions
1. The relationship between the variables $x$ and $y$ is a direct variation.
2. A direct variation can be described by the equation $y = kx$ , where $k$ is the constant of variation.
3. The given point that lies on the direct variation graph is $(4, 2)$ .

To find the constant of variation $k$ , we use the equation of direct variation and the given point.
$y = kx$

Substitute the coordinates of the given point $(4, 2)$ into the direct variation equation.
$2 = k \cdot 4$

Solve for the constant of variation $k$ by dividing both sides of the equation by 4.
$k = \frac{2}{4}$

Simplify the fraction to find the value of $k$ .
$k = \frac{1}{2}$

Now that we have the value of $k$ , we can write the equation of the direct variation.
$y = \frac{1}{2}x$
The equation of the direct variation that includes the given point $(4, 2)$ is $y = \frac{1}{2}x$ , and the constant of variation $\mathrm{k}$ is $\frac{1}{2}$ .