Example

A light green latex paint that is 20% yellow paint is combined with a darker green latex paint that is 45% yellow paint. How many gallons of each paint must be used to create 15 gallons of a green paint that is 25% yellow paint?

Let x be the number of gallons of the 20% yellow paint and let y be the number of gallons of the 40% yellow paint. This means that we want those two numbers to add up to 15: x+y=15.

Now if we want 15 gallons of 25% yellow paint, that means we want 0.25⋅15=3.75 gallons of pure yellow pigment. The expression 0.20⋅x represents the amount of pure yellow pigment in the x gallons of 20% yellow paint. The expression 0.45⋅y represents the amount of pure yellow pigment in the y gallons of 45% yellow paint. Combing the last two adds up to the 3.75 gallons of pure pigment in the final mixture:

0.20x+0.40y=3.75

The system is: \left\{\begin{aligned} x+y &=15 \\ 0.20 x+0.45 y &=3.75 \end{aligned}\right.

We can isolate one variable and use substitution to solve the system:

x=15−y

Now solve for y.

\begin{aligned} 0.20(15-y)+0.45 y &=3.75 & & \\ 3-0.20 y+0.45 y &=3.75 & & \text { Distributive Property } \\ 3+0.2 y &=3.75 & & \text { Add like terms. } \\ 0.25 y &=0.75 & & \text { Subtract } 3 . \\ y &=3 & & \text { Divide by } 0.25 \end{aligned}

Now we can plug in y=3 into x+y=15:

x+y=15 \Rightarrow x+3=15 \Rightarrow x=12

This means 12 gallons of 20% yellow paint should be mixed with 3 gallons of 45% yellow paint in order to get 15 gallons of 25% yellow paint.