# If y varies directly as x and z, and y = 40 when x = 5 and z = 4, find y when x = 2 and z = 1.

## Question:

If y varies directly as x and z, and y = 40 when x = 5 and z = 4, find y when x = 2 and z = 1.

## Joint Variation:

Joint variation gives a relationship between three or more variables whereby the dependent variable varies directly as all the independent variables. For example, if a variable {eq}w {/eq} varies jointly as {eq}\rm x, \; y, \; and\; z {/eq}, the equation relating the four variables can be written as {eq}w = k xyz {/eq}, where {eq}k {/eq} is the variation constsnt.

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

• {eq}y\propto xz {/eq}

Removing the proportinality sign and adding a proportionality constant, we get:

• {eq}y = kxz {/eq}

Given that {eq}y = 40 {/eq} when {eq}x = 5 {/eq} and {eq}z = 4 {/eq}, then:

• {eq}40 = k\times 5\times 4 {/eq}
• {eq}40 = 20k {/eq}

Solving for k:

• {eq}k = \dfrac{40}{20} {/eq}
• {eq}k = 2 {/eq}

Therefore:

• {eq}y = 2xz {/eq}

Using this equation, the value of {eq}y {/eq} when {eq}x = 2 {/eq} and {eq}z = 1 {/eq} is:

• {eq}y = 2\times 2\times 1 {/eq}
• {eq}\boxed{\color{blue}{y = 4}} {/eq}