Exercise 94 Page 616

Let's call the number of gallons Antoine needs of powder blue and spring blue for

x

and

y

respectively. The sum of these quantities should equal

1

gallon. With this information, we can write the following equation.

x + y = 1

We are also told that the mixture should be

4 %

blue. Powder blue and spring blue contains

2 %

and

4 %

blue, respectively. With this information we can write a second equation.

0.02 x + 0.1 y = 0.04 (1) ⇓ 0.02 x + 0.1 y = 0.04

If we combine these equations we get a system of equations, which we can solve by the Substitution Method.

{x + y = 1 0.02 x + 0.1 y = 0.04 (I) (II)

$(I):$ $LHS - y = RHS - y$

{x = 1 - y 0.02 x + 0.1 y = 0.04

$(II):$ $x = 1 - y$

{x = 1 - y 0.02 (1 - y) + 0.1 y = 0.04

▼

(II):

Solve for

y

$(II):$ Distribute $0.02$

{x = 1 - y 0.02 - 0.02 y + 0.1 y = 0.04

$(II):$ Add terms

{x = 1 - y 0.02 + 0.08 y = 0.04

$(II):$ $LHS - 0.02 = RHS - 0.02$

{x = 1 - y 0.08 y = 0.02

$(II):$ $LHS / 0.08 = RHS / 0.08$

{x = 1 - y y = 0.25

$(I):$ $y = 0.25$

{x = 1 - 0.25 y = 0.25

$(I):$ Subtract term

{x = 0.75 y = 0.25

Antoine should use

0.75

gallons of powder blue and

0.25

gallons of spring blue.