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Section 2.3 Algebraic Solution of Systems

Some familiar formulas are useful in writing equations to solve a problem.

Subsection 1. Use the interest formula, \(I=Pr\)

Subsubsection Example

Example 2.19.

You have $5000 to invest for one year. You want to put part of the money into bonds that pay 7% interest, and the rest of the money into stocks that involve some risk but will pay 12% if successful. Now suppose you decide to invest \(x\) dollars in stocks and \(y\) dollars in bonds.

  1. Use the interest formula, \(I=Pr\text{,}\) to write expressions for the interest earned on the bonds and on the stocks.
  2. Write an equation about the amount invested.
  3. Write an equation to say that the total interest earned was $400.
Solution

  1. Stocks: \(I=0.12x;~~~\)Bonds: \(I=0.07y\)

  2. \(\displaystyle x+y=500\)
  3. \(\displaystyle 0.12x+0.07y=400\)

Subsubsection Exercise

Jerry invested $2000, part in a CD at 4% interest and the remainder in a business venture at 9%. After one year, his income from the business venture was $37 more than his income from the CD. Now suppose Jerry invested \(x\) dollars in the CD and \(y\) dollars in the business venture.

  1. Use the interest formula to write expressions for the interest Jerry earned on the CD and the interest he earned on the business venture.
  2. Write an equation about the amount Jerry invested.
  3. Write an equation about the interest Jerry earned.
Answer
  1. \(\displaystyle 0.04x;~~0.09y\)
  2. \(\displaystyle x+y=2000\)
  3. \(\displaystyle 0.09y=37+0.04x\)

Subsection 2. Use the percent formula, \(P=rW\)

Subsubsection Example

Example 2.21.

A chemist wants to produce 45 quarts of a 40% solution of carbolic acid by mixing a 20% solution with a 50% solution. She uses \(x\) quarts of the 20% solution and \(y\) quarts of the 50% solution.

  1. Write an equation about the total amount of solution.
  2. Use the percent formula, \(P=rW\text{,}\) to write expressions about the amount of carbolic acid in each original solution.
  3. How many quarts of carbolic acid are in the mixture?
  4. Write an equation about the amount of carbolic acid.
Solution
  1. \(\displaystyle x+y=45\)

  2. 20% solution: \(0.20x;~~~\) 50% solution: \(0.50y\)

  3. \(\displaystyle P=rW=0.40(45)\)
  4. \(\displaystyle 0.20x+0.50y=0.40(45)\)

Subsubsection Exercise

A pet store owner wants to mix a 12% saltwater solution and a 30% saltwater solution to obtain 90 liters of a 24% solution. He uses \(x\) quarts of the 12% solution and \(y\) quarts of the 30% solution.

  1. Write an equation about the total amount of saltwater.
  2. Use the percent formula to write expressions about the amount of salt in each original solution.
  3. How many liters of salt are in the mixture?
  4. Write an equation about the amount of salt.
Answer
  1. \(\displaystyle x+y=90\)
  2. \(\displaystyle 0.20x;~~0.30y\)
  3. \(\displaystyle 0.24(90)\)
  4. \(\displaystyle 0.20x+0.30y=0.24(90)\)

Subsection 3. Use the distance formula, \(d=rt\)

Subsubsection Example

Example 2.23.

A river steamer requires 3 hours to travel 24 miles upstream and 2 hours for the return trip downstream. Let \(x\) be the speed of the current and \(y\) the speed of the steamer in still water.

  1. Write an equation about the upstream trip.
  2. Write an equation about the downstream trip.
Solution
  1. The speed of the steamer aginst the current is \(r=y-x\text{,}\) so \(~3(y-x)=24\)
  2. The speed of the steamer with the current is \(r=y+x\text{,}\) so \(~2(y+x)=24\)

Subsubsection Exercise

A yacht leaves San Diego and heads south, traveling at 25 miles per hour. Six hours later a Coast Guard cutter leaves San Diego traveling at 40 miles per hour and pursues the yacht. Let \(x\) be the time it takes the cutter to catch the yacht, and \(y\) the distance it traveled.

  1. Write an equation about the yacht's journey.
  2. Write an equation about the cutter's journey.
Answer
  1. \(\displaystyle 25(x+6)=y\)
  2. \(\displaystyle 40x=y\)