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Section 8.1 Systems of Linear Equations in Two Variables

Subsection 1. Identify the solution of a system

Recall that a solution to a system makes each equation in the system true.

Subsubsection Examples

Example 8.1.

Decide whether \((3,-2)\) is a solution of the system

\begin{gather*} x = 5y+13\\ 2x+3y=0 \end{gather*}
Solution

A solution must satisfy both equations. We substitute \(x=\alert{3}\) and \(y=\alert{-2}\) into the equations.

\begin{align*} \alert{3} = 5(\alert{-2})+13 ? \amp \amp \text{Yes}\\ 2(\alert{3})+3(\alert{-2})=0 ? \amp \amp \text{Yes} \end{align*}

Yes, \((3,-2)\) is a solution

Example 8.2.

Find the solution of the system graphed below.

Linear system
Solution

The solution must lie on both graphs, so it is the intersection point, \(P\text{.}\) The coordinates of point \(P\) are \((50, 1300)\text{,}\) so the solution of the system is \(t=50,~y=1300\text{.}\)

Subsubsection Exercises

Decide whether \((-3,-2)\) is a solution of the system

\begin{align*} x + 3y \amp = -9\\ 3x+2y \amp = -5 \end{align*}
Answer

No

Find the solution of the system graphed below.

Linear system
Answer
\((300, 24)\)

Subsection 2. Write equations in two variables

Applied problems that involve more than one unknown are often easier to model and solve with a system of equations.

Subsubsection Examples

Example 8.5.

Write equations about the number of tables and the number of chairs:

  1. There are four chairs for each table.
  2. Chairs cost $125 each; a table costs $350. Darryl spent $10,200 on tables and chairs.
Solution

Let \(x\) be the number of tables and \(y\) the number of chairs.

  1. The number of chairs is 4 times the number of tables: \(~y=4x\text{.}\)
  2. \(\displaystyle 125y+350x=10,200\)
Example 8.6.

Write equations about the dimensions of a rectangle:

  1. The perimeter of the rectangle is 42 meters.
  2. The length is 3 meters more than twice the width.
Solution

Let \(x\) be the width of the rectangle and \(y\) its length.

  1. \(\displaystyle 2x+2y=42\)
  2. \(\displaystyle y=3+2x\)

Subsubsection Exercises

Write equations about the number of calories in a hamburger and in a chocolate shake.

  1. A hamburger and a chocolate shake together contain 1020 calories.
  2. Two shakes and three hamburgers contain 2710 calories.
Answer
  1. \(\displaystyle x+y=1020\)
  2. \(\displaystyle 3x+2y=2710\)

Write equations about the vertex angle and the base angles of an isosceles triangle.

  1. The vertex angle is \(15 \degree\) less than each base angle.
  2. The sum of the angles in a triangle is \(180 \degree\text{.}\)
Answer
  1. \(\displaystyle y=x-5\)
  2. \(\displaystyle 2x+y=180\)

Subsection 3. Use formulas

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

Subsubsection Example

Example 8.9.

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. \(\displaystyle 0.12x+0.07y=400\)
Example 8.10.

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. \(\displaystyle 0.20x+0.50y=0.40(45)\)
Example 8.11.

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 Exercises

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. \(\displaystyle 0.09y=37+0.04x\)

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. \(\displaystyle 0.20x+0.30y=0.24(90)\)

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. \(\displaystyle 40x=y\)