y varies directly with x. If x=52 when y=130, find x when y=140.

Question:

y varies directly with x. If x=52 when y=130, find x when y=140.

Direct Variation:

Direct variation explains a relationship between variables that affect each other in the same direction. That is, when an increase/decrease in one variable leads to an increase/decrease in the other variable.

If {eq}y {/eq} varies directly with {eq}x {/eq}, we can write this as:

• {eq}y\propto x {/eq}

Removing the proportionality sign, we have:

• {eq}y = k x {/eq}

Where {eq}k {/eq} is the constant of variation.

Given that {eq}x = 52 {/eq} when {eq}y = 130 {/eq}, the proportionality constant is equal to:

• {eq}130 = k \times 52 {/eq}
• {eq}k = \dfrac{130}{52} = 2.5 {/eq}

Thus, the function explaining the relationship between the two variables is:

• {eq}y = 2.5x {/eq}

Using the above function, we can determine the value of y for any given value of x. Thus, the value of x when y = 140 is equal to:

• {eq}140 = 2.5x {/eq}
• {eq}x = \dfrac{140}{2.5} = \boxed{56} {/eq}