Math  /  Algebra

QuestionIt is given that yy is directly proportional to xx. \begin{tabular}{|c|c|} \hline x\boldsymbol{x} & y\boldsymbol{y} \\ \hline 1 & -4 \\ \hline 2 & -8 \\ \hline 3 & -12 \\ \hline 4 & -16 \\ \hline 5 & -20 \\ \hline \end{tabular}
What is the constant of variation? \square

Studdy Solution

STEP 1

What is this asking? If yy changes proportionally to xx, what's the multiplier between them? Watch out! Don't mix up direct and inverse proportionality!
Direct means they both go up or down together.

STEP 2

1. Define direct proportionality
2. Find the constant of variation

STEP 3

When two things are *directly proportional*, it means if one gets bigger, the other gets bigger by the same factor.
Like, if you buy twice as much candy, you pay twice as much money (unless you get a bulk discount, but we'll assume no discounts here!). Mathematically, we can write this as y=kxy = k \cdot x, where kk is our **constant of variation** – the magic multiplier!

STEP 4

We're given a handy table of xx and yy values.
Let's pick *any* pair and plug them into our equation y=kxy = k \cdot x to find kk.
Let's choose the first pair: x=1x = 1 and y=4y = -4.

STEP 5

Substituting the values, we get 4=k1-4 = k \cdot 1.

STEP 6

To solve for kk, we **divide both sides** of the equation by 11.
We divide by 11 because we want to isolate kk by dividing to one.
This gives us k=41=4k = \frac{-4}{1} = -4.
So, our **constant of variation** is 4-4!

STEP 7

Let's **double-check** with another pair from the table, just to be super sure.
Let's take x=2x = 2 and y=8y = -8.
Plugging these into y=kxy = k \cdot x, we get 8=k2-8 = k \cdot 2.

STEP 8

**Dividing both sides** by 22 to isolate kk, we get k=82=4k = \frac{-8}{2} = -4.
We divide by 22 because we want to isolate kk by dividing to one.
Phew, we got the same answer!
This confirms our **constant of variation** is indeed 4-4.

STEP 9

The constant of variation is 4-4.

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