Math

Question Determine if the linear equation 3x+6y=123x + 6y = 12 represents a function.

Studdy Solution

STEP 1

Assumptions
1. We are given an equation in two variables, xx and yy.
2. We need to determine if this equation represents a function of xx.

STEP 2

Recall the definition of a function. A function ff from a set AA to a set BB is a rule that assigns to each element xx in AA exactly one element yy in BB.

STEP 3

To determine if the given equation represents yy as a function of xx, we must be able to solve for yy in terms of xx such that for each xx value there is exactly one yy value.

STEP 4

Solve the given equation for yy in terms of xx.
3x+6y=123x + 6y = 12

STEP 5

Subtract 3x3x from both sides of the equation to isolate terms with yy on one side.
6y=123x6y = 12 - 3x

STEP 6

Divide both sides of the equation by 66 to solve for yy.
y=123x6y = \frac{12 - 3x}{6}

STEP 7

Simplify the right-hand side of the equation.
y=212xy = 2 - \frac{1}{2}x

STEP 8

Now that we have yy expressed as a function of xx, we can see that for each value of xx, there is exactly one corresponding value of yy. This is because the equation is linear and there are no restrictions on the values of xx that would cause there to be more than one value of yy for any given xx.

STEP 9

Conclude that the given equation 3x+6y=123x + 6y = 12 does represent yy as a function of xx.
The equation represents a function of xx.

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