7.1: Linear Systems by Graphing
In a previous chapter, you learned that 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.
Example 1: Determine which of the points (1, 3), (0, 2), or (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*}
Solution: A solution to a system is an ordered pair that is in common to all the algebraic sentences. 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. \ Yes, \ this \ ordered \ pairs \ checks. \\ 3 = 2(1)+3; \ 3=5. \ No, \ this \ ordered \ pair \ does \ not \ check. \end{cases}\end{align*}
Check (0, 2) : \begin{align*} \begin{cases} 2=4(0)-1; \ 2=-1. \ No, \ this \ ordered \ pair \ does \ not \ check. \\ 2=2(0)+3; \ 2=3. \ No, \ this \ ordered \ pair \ does \ not \ check. \end{cases}\end{align*}
Check (2, 7) : \begin{align*} \begin{cases} 7=4(2)-1; \ 7=7. \ Yes, \ this \ ordered \ pairs \ checks. \\ 7=2(2)+3; \ 7=7. \ Yes, \ this \ ordered \ pairs \ 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*}
Example: Find the solution to the system \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 in Chapter 4.
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; \quad 1=1\\ 1=-2(2)+5; \ 1=1 \end {cases}\end{align*}
The coordinate checks in both sentences. Therefore, (2, 1) is a solution to the system \begin{align*}\begin{cases} y=3x-5\\ y=-2x+5 \end{cases}\end{align*}.
Example 2: Solve the system \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).
\begin{align*}\begin{cases} -1+3=2; \ 2=2\\ \qquad 3=3 \end{cases}\end{align*}
The coordinate is in common to each sentence and 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.
Using the system from the above example, \begin{align*}\begin{cases} y=3x-5\\ y=-2x+5 \end{cases}\end{align*}, we will use the 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.
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 \begin{align*}distance=rate \times 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.
Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both.
CK-12 Basic Algebra: Solving Linear Systems by Graphing (8:30)
- 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 in 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*}y=x+3\!\\ y =-x+3\end{align*}
- \begin{align*}y=3x-6\!\\ y =-x+6\end{align*}
- \begin{align*}2x=4\!\\ y =-3\end{align*}
- \begin{align*}y=-x+5\!\\ -x+y =1\end{align*}
- \begin{align*}x+2y=8\!\\ 5x+2y =0\end{align*}
- \begin{align*}3x+2y=12\!\\ 4x-y =5\end{align*}
- \begin{align*}5x+2y=-4\!\\ x-y = 2\end{align*}
- \begin{align*}2x+4=3y\!\\ x-2y+4=0\end{align*}
- \begin{align*}y=\frac{x}{2}-3\!\\ 2x-5y = 5\end{align*}
- \begin{align*}y=4\!\\ x =8-3y\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 when the total cost of repair 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?
Mixed Review
- Solve for \begin{align*}h: 25 \ge |2h+5|\end{align*}.
- Subtract \begin{align*}\frac{4}{3}-\frac{1}{2}\end{align*}.
- You write the letters to ILLINOIS on separate pieces of paper and place them into a hat.
- Find \begin{align*}P\end{align*}(drawing an \begin{align*}I\end{align*}).
- Find the odds for drawing an \begin{align*}L\end{align*}.
- Graph \begin{align*}x<2\end{align*} on a number line and on a Cartesian plane.
- Give an example of an ordered pair in quadrant II.
- The data below show the average life expectancy in the United States for various years.
- Use the method of interpolation to find the average life expectancy in 1943.
- Use the method of extrapolation to find the average life expectancy in 2000.
- Find an equation for the line of best fit. How do the predictions of this model compare to your answers in questions a) and b)?
Birth Year | Female | Male | Combined |
---|---|---|---|
1940 | 65.2 | 60.8 | 62.9 |
1950 | 71.1 | 65.6 | 68.2 |
1960 | 73.1 | 66.6 | 69.7 |
1970 | 74.7 | 67.1 | 70.8 |
1975 | 76.6 | 68.6 | 72.6 |
1980 | 77.5 | 70.0 | 73.7 |
1985 | 78.2 | 71.2 | 74.7 |
1990 | 78.8 | 71.8 | 75.4 |
1995 | 78.9 | 72.5 | 75.8 |
1998 | 79.4 | 73.9 | 76.7 |