Suppose that on a test there are multiple choice questions and fill-in-the-blank questions. Julia answered 8 multiple choice questions and 3 fill-in-the-blank questions correctly, while Jason answered 6 multiple choice questions and 5 fill-in-the-blank questions correctly. If Julia got a total of 14 points and Jason got 16 points, how many points is each type of question worth? Could you use a graph to solve the system of linear equations representing this scenario?

### Graphing Linear Systems

The intersection of two sets is joined by the word “and.” This word also joins two or more equations or inequalities. A set of algebraic sentences joined by the word “and” is called a **system.**

The **solution(s)** to a system is the set of ordered pairs that is in common to each algebraic sentence.

#### Let's determine which of the points (1, 3), (0, 2), and (2, 7) is a solution to the following system of equations:

#### \begin{align*} \begin{cases} y = 4x - 1 \\ y = 2x + 3 \end{cases}\end{align*}

A solution to a system is an ordered pair that is a solution to all the algebraic statements in the system. To determine if a particular ordered pair is a solution, substitute the coordinates for the variables \begin{align*}x\end{align*} and \begin{align*}y\end{align*} in each sentence and check.

Check (1, 3) : \begin{align*} \begin{cases} 3 = 4(1)-1; \ 3=3. \text{ Yes, this ordered pair checks.} \\ 3 = 2(1)+3; \ 3=5. \text{ No, this ordered pair does not check.} \end{cases}\end{align*}

Check (0, 2) : \begin{align*} \begin{cases} 2=4(0)-1; \ 2=\text{-}1. \text{ No, this ordered pair does not check.} \\ 2=2(0)+3; \ 2=3. \ \text{ No, this ordered pair does not check.} \end{cases}\end{align*}

Check (2, 7) : \begin{align*} \begin{cases} 7=4(2)-1; \ 7=7. \text{ Yes, this ordered pair checks.}\\ 7=2(2)+3; \ 7=7. \text{ Yes, this ordered pair checks.} \end{cases}\end{align*}

Because the coordinate (2, 7) works in both equations simultaneously, it is a solution to the system.

To determine the coordinate that is in common to each sentence in the system, each equation can be graphed. The point at which the lines **intersect** represents the solution to the system. The solution can be written two ways:

- As an ordered pair, such as (2, 7)
- By writing the value of each variable, such as \begin{align*}x=2, \ y=7\end{align*}.

#### Now, let's find the solution to the following by graphing:

\begin{align*}\begin{cases} y=3x-5\\ y=-2x+5 \end {cases}\end{align*}

By graphing each equation and finding the point of intersection, you find the solution to the system.

Each equation is written in slope-intercept form and can be graphed accordingly.

The lines appear to intersect at the ordered pair (2, 1). Is this the solution to the system?

\begin{align*}\begin{cases} 1=3(2)-5; & 1=1\\ 1=-2(2)+5; & 1=1 \end {cases}\end{align*}

The coordinates check in both sentences. Therefore, (2, 1) is a solution to the system.

The greatest strength of the graphing method is that it offers a very visual representation of a system of equations and its solution. You can see, however, that determining a solution from a graph would require very careful graphing of the lines and is really practical only when you are certain that the solution gives integer values for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}. In most cases, this method can offer only approximate solutions to systems of equations. For exact solutions, other methods are necessary.

#### Solving Systems Using a Graphing Calculator

A graphing calculator can be used to find or check solutions to a system of equations. To solve a system graphically, you must graph the two lines on the same coordinate axes and find the point of intersection. You can use a graphing calculator to graph the lines as an alternative to graphing the equations by hand.

#### Let's use a graphing calculator to solve the problem:

Using the system from the above example, \begin{align*}\begin{cases} y=3x-5\\ y=-2x+5 \end{cases}\end{align*} use a graphing calculator to find the approximate solutions to the system.

Begin by entering the equations into the \begin{align*}Y=\end{align*} menu of the calculator.

You already know the solution to the system is (2, 1). The window needs to be adjusted so an accurate picture is seen. Change your window to the **default window**.

See the graphs by pressing the **GRAPH** button.

The solution to a system is the intersection of the equations. To find the intersection using a graphing calculator, locate the **Calculate** menu by pressing \begin{align*}2^{nd}\end{align*} and **TRACE**. Choose option #5 – **INTERSECTION**.

The calculator will ask you **“First Curve?”** Hit **ENTER**. The calculator will automatically jump to the other curve and ask you **“Second Curve?”** Hit **ENTER**. The calculator will ask, **“Guess?”** Hit **ENTER**. The intersection will appear at the bottom of the screen.

#### Now, let's solve the following real-world problem:

Peter and Nadia like to race each other. Peter can run at a speed of 5 feet per second and Nadia can run at a speed of 6 feet per second. To be a good sport, Nadia likes to give Peter a head start of 20 feet. How long does Nadia take to catch up with Peter? At what distance from the start does Nadia catch up with Peter?

Begin by translating each runner’s situation into an algebraic sentence using \begin{align*}\text{distance}= \text{rate} \times \text{time}\end{align*}.

Peter: \begin{align*}d=5t+20\end{align*}

Nadia: \begin{align*}d=6t\end{align*}

The question asks when Nadia catches Peter. The solution is the point of intersection of the two lines. Graph each equation and find the intersection.

The two lines cross at the coordinate \begin{align*}t=20, \ d=120\end{align*}. This means after 20 seconds Nadia will catch Peter. At this time, they will be at a distance of 120 feet. Any time after 20 seconds Nadia will be farther from the starting line than Peter.

### Examples

#### Example 1

Earlier, you were told that Julia answered 8 multiple choice questions and 3 fill-in-the-blank questions correctly while Jason answered 6 multiple choice questions and 5 fill-in-the-blank questions correctly. If Julia got a total of 14 points and Jason got 16 points, how many points is each type of question worth?

Let

The graph of the equations is shown below:

The intersection looks like the point (1, 2). Let's check the answer:

\begin{align*}\begin{cases} 8(1) + 3(2) = 14; &8+6 = 14 14 = 14\\ 6(1) + 5(2) = 16; &6 + 10= 16; 16 = 16\end {cases}\end{align*}

The multiple choice questions are worth 1 point and the fill-in-the-blank questions are worth 2 points.

#### Example 2

Solve the system \begin{align*}\begin{cases} x+y=2& \\ \ \, \quad y=3 \end{cases}\end{align*}

The first equation is written in standard form. Using its intercepts will be the easiest way to graph this line.

The second equation is a horizontal line three units up from the origin.

The lines appear to intersect at (–1, 3).

\begin{align*}\begin{cases} -1+3=2; \ & 2=2\\ &3=3 \end{cases}\end{align*}

The coordinates are a solution to each sentence and are a solution to the system.

### Review

- Define a system.
- What is the solution to a system?
- Explain the process of solving a system by graphing.
- What is one problem with using a graph to solve a system?
- What are the two main ways to write the solution to a system of equations?
- Suppose Horatio says the solution to a system is (4, –6). What does this mean visually?
- Where is the
**“Intersection”**command located on your graphing calculator? What does it do? - In the race example, who is farther from the starting line at 19.99 seconds? At 20.002 seconds?

Determine which ordered pair satisfies the system of linear equations.

- \begin{align*}\begin{cases} y=3x-2\\ y=-x\end{cases}\end{align*} \begin{align*}(1,4), \ (2,9), \ \left ( \frac{1}{2}, \ -\frac{1}{2} \right )\end{align*}
- \begin{align*}\begin{cases} y=2x-3\\ y=x+5\end{cases}\end{align*} \begin{align*}(8,13), \ (-7,6), \ (0,4)\end{align*}
- \begin{align*}\begin{cases} 2x+y=8\\ 5x+2y=10\end{cases}\end{align*} \begin{align*}(-9,1), \ (-6,20), \ (14,2)\end{align*}
- \begin{align*}\begin{cases} 3x+2y=6\\ y=\frac{x}{2}-3\end{cases}\end{align*} \begin{align*}\left ( 3, -\frac{3}{2} \right ), \ (-4,3), \ \left ( \frac{1}{2}, 4\right )\end{align*}

In 13 – 22, solve the following systems by graphing.

- \begin{align*}\begin {cases} y=x+3\\ y =-x+3 \end{cases}\end{align*}
- \begin{align*}\begin{cases} y=3x-6\\ y =-x+6 \end{cases}\end{align*}
- \begin{align*}\begin{cases} 2x=4\\ y =-3 \end{cases}\end{align*}
- \begin{align*}\begin{cases}y=\text{-}x+5\\ \text{-}x+y =1 \end{cases}\end{align*}
- \begin{align*}\begin{cases}x+2y=8\\ 5x+2y =0 \end{cases}\end{align*}
- \begin{align*}\begin{cases}3x+2y=12\\ 4x-y =5 \end{cases}\end{align*}
- \begin{align*}\begin{cases}5x+2y=-4\\ x-y = 2 \end{cases}\end{align*}
- \begin{align*}\begin{cases}2x+4=3y\\ x-2y+4=0 \end{cases}\end{align*}
- \begin{align*}\begin{cases}y=\frac{x}{2}-3\\ 2x-5y = 5 \end{cases}\end{align*}
- \begin{align*}\begin{cases} y=4\\ x =8-3y \end{cases}\end{align*}
- Mary’s car is 10 years old and has a problem. The repair man indicates that it will cost her $1200.00 to repair her car. She can purchase a different, more efficient car for $4,500.00. Her present car averages about $2,000.00 per year for gas while the new car would average about $1,500.00 per year. Find the number of years for which the total cost of repairs would equal the total cost of replacement.
- Juan is considering two cell phone plans. The first company charges $120.00 for the phone and $30 per month for the calling plan that Juan wants. The second company charges $40.00 for the same phone, but charges $45 per month for the calling plan that Juan wants. After how many months would the total cost of the two plans be the same?
- A tortoise and hare decide to race 30 feet. The hare, being much faster, decided to give the tortoise a head start of 20 feet. The tortoise runs at 0.5 feet/sec and the hare runs at 5.5 feet per second. How long will it be until the hare catches the tortoise?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 7.1.