Skip to main content

Section 2.2 Linear Systems

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 2.7.

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 2.8.

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. Use the formula for profit, \(P=R-C\)

Profit.

To find the profit earned by a company we subtract the costs from the revenue.

\begin{equation*} \text{Profit} = \text{Revenue} - \text{Cost}~~~~~~~~P=R-C \end{equation*}

"Revenue" is the amount of money a company takes in from selling its product. A negative profit is the same as a loss.

Subsubsection Examples

Example 2.11.

The owner of a sandwich shop spent $800 last week for labor and supplies. She received $1150 in revenue. What was her profit?

Solution

We evaluate the formula with \(R=1150\) and \(C=800\) to find

\begin{align*} P \amp = R-C\\ \amp = 1150 - 800 = 350 \end{align*}

The owner's profit was $350.

Example 2.12.

EcoGreen made $1848 profit on low-flow shower heads last year, and spent $3426 in costs. What was their revenue from shower heads?

Solution

We usse the profit formula and solve the equation

\begin{equation*} 1848 = R - 3246 \end{equation*}

to find that \(R=5094\text{.}\) Their revenue was $5094.

Subsubsection Exercises

  1. The Earth Alliance made $6000 in revenue from selling tickets to Earth Day, an educational event for children. Write an expression for their profit in terms of their costs.
  2. What was their profit if their costs were $2500?
Answer
  1. \(\displaystyle P=6000-C \)
  2. $3500

Last week Moe's Auto Shop took in $5400 in revenue, but experienced a loss of $800. What were Moe's costs last week?

Answer
$6200

Subsection 3. 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 2.15.

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 2.16.

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. \(\displaystyle 2x+y=180\)