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Write a direct variation equation to relate to x and y, and solve. If y=-16 when x=4, find x when y=20.

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2 Answers

The direct variation is "y=kx" . As "x" increases the value of "y" increases in constant rate.
In our case, we have to find the constant rate (coefficient k by x). Let's replace "x" and "y" in formula of direct variation by given values: -16=k(4) and solve this equation for "k".
We have to divide both sides of equation by 4 ,
-16/4 = k(4/4)
-4=k
So, the direct variation equation will be
y = -4x
Now let's find x , when y=20 . We have to replace "y" by 20 in above equation and solve it for "x"
20 = -4x
20/(-4) = (-4)/(-4)x
x = (-5) .

y = -4x

When x = 4, y = -16

When y = 20, x = -5

 

Explanation:

First off, definitions: Direct variations refers to two variables being related with neither being the denominator of a fraction. (nothing along the lines of y = 1/x)

So, given that, I simply found what the ratio was between the two values. Something like buying gas at the gas station. The ratio between the gas you get and the money you pay is the price per gallon of gas. In this case, the proportion is -4 (-16 divided by 4).

Thus we have a relationship. For each y, you have negative four x's. Thus, y = -4x.

Finding the value of x when y = 20 simply requires plugging values in. Remember, a fundamental principle in all of Algebra is substitution. If two things are equal, they are interchangeable. So, if you saw "3 + 4" in a mathematical expression, you could simply erase it and write 7, so long as no other mathematical rules (multiplication before addition, for instance) interferes.

Thus, if y = 20, we can take our original equation and replace y with 20.

y = -4x       =>     20 = -4x

Then, simply solve for x, keeping in mind that when two items in math are written against each other, it means multiplication. Thus, we reverse the multiplication to division and divide both sides of the equation by -4.

20 / -4 = -5