# Graphs of Linear Systems

## Graph lines to identify intersection points

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Graphing Linear Systems

4. ### What is the primary way  to write the solution to a system of equations?

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. In this Concept, you'll learn to solve linear systems by graphing so that you can answer questions like this.

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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.

Credit: CK-12 Foundation

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Example A

Find the solution to the system.

{y=3x5y=2x+5\begin{align*}\begin{cases} y=3x-5\\ y=-2x+5 \end {cases}\end{align*}.

### Solution

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 using the methods learned previously.

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

{1=3(2)5;1=11=2(2)+5; 1=1\begin{align*}\begin{cases} 1=3(2)-5; \quad 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 {y=3x5y=2x+5\begin{align*}\begin{cases} y=3x-5\\ y=-2x+5 \end{cases}\end{align*}.

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 x\begin{align*}x\end{align*} and y\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.

### Video Example!!

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### Real World Example!!

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?

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

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

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

The two lines cross at the coordinate t=20, d=120\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.

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### Guided Practice

Solve the system {x+y=2y=3\begin{align*}\begin{cases} x+y=2\\ \qquad y=3 \end{cases}\end{align*}.

Solution: 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).

{1+3=2; 2=23=3\begin{align*}\begin{cases} -1+3=2; \ 2=2\\ \qquad 3=3 \end{cases}\end{align*}

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

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### Practice

1. Define a system.
2. What is the solution to a system?
3. Explain the process of solving a system by graphing.
4. What is one problem with using a graph to solve a system?
5. What are the two main ways to write the solution to a system of equations?
6. Suppose Horatio says the solution to a system is (4, –6). What does this mean visually?
7. Where is the “Intersection” command located on your graphing calculator? What does it do?
8. 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.

1. {y=3x2y=x\begin{align*}\begin{cases} y=3x-2\\ y=-x\end{cases}\end{align*} ;(1,4), (2,9), (12, 12)\begin{align*};(1,4), \ (2,9), \ \left ( \frac{1}{2}, \ -\frac{1}{2} \right )\end{align*}
2. {y=2x3y=x+5\begin{align*}\begin{cases} y=2x-3\\ y=x+5\end{cases}\end{align*} ;(8,13), (7,6), (0,4)\begin{align*};(8,13), \ (-7,6), \ (0,4)\end{align*}
3. {2x+y=85x+2y=10\begin{align*}\begin{cases} 2x+y=8\\ 5x+2y=10\end{cases}\end{align*} ;(9,1), (6,20), (14,2)\begin{align*}; (-9,1), \ (-6,20), \ (14,2)\end{align*}
4. {3x+2y=6y=x23\begin{align*}\begin{cases} 3x+2y=6\\ y=\frac{x}{2}-3\end{cases}\end{align*} ;(3,32), (4,3), (12,4)\begin{align*}; \left ( 3, -\frac{3}{2} \right ), \ (-4,3), \ \left ( \frac{1}{2}, 4\right )\end{align*}
1. 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.
2. 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?
3. 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?

Credit: CK-12 Foundation

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