7.4: Linear Systems with Addition or Subtraction
Suppose two extended families went to an amusement park, where adult tickets and children's tickets have different prices. If the first family had 5 adults and 8 children and paid a total of $124, and the second family had 5 adults and 12 children and paid $156, how much did each type of ticket cost? Could you write a system of equations representing this situation? If you wanted to solve the system by elimination, how would you go about doing it? In this Concept, you'll learn how to use elimination to solve a system of linear equations similar to the one representing this scenario by addition or subtraction.
Watch This
Multimedia Link: For help with solving systems by elimination, visit this site: http://www.teachertube.com/viewVideo.php?title=Solving_System_of_Equations_using_Elimination&video_id=10148 Teacher Tube video.
Guidance
As you noticed in the previous Concept, solving a system algebraically will give you the most accurate answer and in some cases, it is easier than graphing. However, you also noticed that it took some work in several cases to rewrite one equation before you could use the Substitution Property. There is another method used to solve systems algebraically: the elimination method.
The purpose of the elimination method to solve a system is to cancel, or eliminate, a variable by either adding or subtracting the two equations. This method works well if both equations are in standard form.
Example A
If one apple plus one banana costs $1.25 and one apple plus two bananas costs $2.00, how much does it cost for one banana? One apple?
Solution: Begin by defining the variables of the situation. Let \begin{align*}a=\end{align*}
\begin{align*}\begin{cases}
a+b=1.25\\
a+2b=2.00 \end{cases}\end{align*}
You could rewrite the first equation and use the Substitution Property here, but because both equations are in standard form, you can also use the elimination method.
Notice that each equation has the value \begin{align*}1a\end{align*}
\begin{align*}& \qquad a + b \ =1.25\\
&\underline{\;\;\; (a+2b=2.00)\;\;\;}\\
& \qquad \quad b =0.75\\
& \qquad \qquad \ b =0.75\end{align*}
Therefore, one banana costs $0.75, or 75 cents. By subtracting the two equations, we were able to eliminate a variable and solve for the one remaining.
How much is one apple? Use the first equation and the Substitution Property.
\begin{align*}a+0.75&=1.25\\
a&=0.50 \rightarrow one \ apple \ costs \ 50 \ cents\end{align*}
Example B
Solve the system \begin{align*}\begin{cases}
3x+2y=11\\
5x2y=13\end{cases}\end{align*}
Solution: These equations would take much more work to rewrite in slopeintercept form to graph, or to use the Substitution Property. This tells us to try to eliminate a variable. The coefficients of the \begin{align*}x\end{align*}
Looking at the \begin{align*}y\end{align*}
\begin{align*}& \qquad \ 3x+2y =11\\
&\underline{\;\; + \ (5x2y) =13 \;\;}\\
& \qquad \ 8x+0y =24\end{align*}
The resulting equation is \begin{align*}8x=24\end{align*}
\begin{align*}3(3)+2y&=11\\
9+2y&=11\\
2y&=2\\
y&=1\end{align*}
The point of intersection of these two equations is (3, 1).
Example C
Andrew is paddling his canoe down a fastmoving river. Paddling downstream he travels at 7 miles per hour, relative to the river bank. Paddling upstream, he moves slower, traveling at 1.5 miles per hour. If he paddles equally hard in both directions, calculate, in miles per hour, the speed of the river and the speed Andrew would travel in calm water.
Solution: We have two unknowns to solve for, so we will call the speed that Andrew paddles at \begin{align*}x\end{align*}
\begin{align*}\text{Downstream Equation} && x+y&=7\\
\text{Upstream Equation} && xy&=1.5\end{align*}
Notice \begin{align*}y\end{align*}
\begin{align*}& \qquad \ x+y=7\\
&\underline{\;\; + \ (xy)=1.5 \;\;}\\
& \quad \ 2x+0y=8.5\\
& \qquad \quad \ \ 2x=8.5\end{align*}
Therefore, \begin{align*}x=4.25\end{align*}
\begin{align*}4.25y&=1.5\\
y&=2.75\\
y&=2.75\end{align*}
The stream’s current is moving at a rate of \begin{align*}2.75 \ miles/hour\end{align*}
Video Review
Guided Practice
Solve the system \begin{align*}\begin{cases}
5s+2t=6\\
9s+2t=22\end{cases}\end{align*}
Solution:
Since these equations are both written in standard form, and both have the term \begin{align*}2t\end{align*}
\begin{align*}& \qquad \ 5s+2t=6\\
&\underline{\;\;  \ (9s+2t = 22) \;\;}\\
& \qquad \ 4s+0t =16\\
& \qquad \ 4s=16\\
& \qquad \ s=4\end{align*}
\begin{align*}5(4)+2t&=6\\
20+2t&=6\\
2t&=14\\
t&=7\end{align*}
The solution is \begin{align*}(4,7)\end{align*}
Practice
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. CK12 Basic Algebra: Solving Linear Systems by Elimination (12:44)
 What is the purpose of the elimination method to solve a system? When is this method appropriate?
In 2 – 10, solve each system using elimination.

\begin{align*}\begin{cases}
2x+y=17\\
8x3y=19 \end{cases}\end{align*}
{2x+y=−178x−3y=−19 
\begin{align*}\begin{cases}
x+4y=9\\
2x5y=12 \end{cases}\end{align*}
{x+4y=−9−2x−5y=12 
\begin{align*}\begin{cases}
2x5y=10\\
x+4y=8 \end{cases}\end{align*}
{−2x−5y=−10x+4y=8  \begin{align*}\begin{cases} x3y=10\\ 8x+5y=15 \end{cases}\end{align*}
 \begin{align*}\begin{cases} x6y=18\\ x6y=6 \end{cases}\end{align*}
 \begin{align*}\begin{cases} 5x3y=14\\ x3y=2 \end{cases}\end{align*}
 \begin{align*}&3x+4y=2.5\\ &5x4y=25.5 \end{align*}
 \begin{align*}&5x+7y=31\\ &5x9y=17 \end{align*}
 \begin{align*}&3y4x=33\\ &5x3y=40.5 \end{align*}
 Nadia and Peter visit the candy store. Nadia buys three candy bars and four fruit rollups for $2.84. Peter also buys three candy bars, but he can afford only one fruit rollup. His purchase costs $1.79. What is the cost of each candy bar and each fruit rollup?
 A small plane flies from Los Angeles to Denver with a tail wind (the wind blows in the same direction as the plane), and an airtraffic controller reads its groundspeed (speed measured relative to the ground) at 275 miles per hour. Another identical plane moving in the opposite direction has a groundspeed of 227 miles per hour. Assuming both planes are flying with identical airspeeds, calculate the speed of the wind.
 An airport taxi firm charges a pickup fee, plus an additional permile fee for any rides taken. If a 12mile journey costs $14.29 and a 17mile journey costs $19.91, calculate:
 the pickup fee
 the permile rate
 the cost of a sevenmile trip
 Calls from a callbox are charged per minute at one rate for the first five minutes, and then at a different rate for each additional minute. If a sevenminute call costs $4.25 and a 12minute call costs $5.50, find each rate.
 A plumber and a builder were employed to fit a new bath, each working a different number of hours. The plumber earns $35 per hour, and the builder earns $28 per hour. Together they were paid $330.75, but the plumber earned $106.75 more than the builder. How many hours did each work?
 Paul has a parttime job selling computers at a local electronics store. He earns a fixed hourly wage, but he can earn a bonus by selling warranties for the computers he sells. He works 20 hours per week. In his first week, he sold eight warranties and earned $220. In his second week, he managed to sell 13 warranties and earned $280. What is Paul’s hourly rate, and how much extra does he get for selling each warranty?
Mixed Review
 Baxter the golden retriever is lying in the sun. He casts a shadow of 3 feet. The doghouse he is next to is 3 feet tall and casts an 8foot shadow. What is Baxter's height?
 A botanist watched the growth of a lily. At 3 weeks, the lily was 4 inches tall. Four weeks later, the lily was 21 inches tall. Assuming this relationship is linear:
 Write an equation to show the growth pattern of this plant.
 How tall was the lily at the 5.5week mark?
 Is there a restriction on how high the plant will grow? Does your equation show this?
 The “Wave” is an exciting pasttime at football games. To prepare, students in a math class took the data in the table below.
 Find a linear regression equation for this data. Use this model to estimate the number of seconds it will take for 18 students to complete a round of the wave.
 Use the method of interpolation to determine the amount of time it would take 18 students to complete the wave.
\begin{align*}& s \ (\text{number of students in wave})&& 4 && 8 && 12 && 16 && 20 && 24 && 28 && 30\\ & t \ (\text{time in seconds to complete on full round}) && 2 && 3.2 && 4 && 5.6 && 7 && 7.9 && 8.6 && 9.1\end{align*}
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Here you'll learn how to use the elimination method in cases of simple addition or subtraction to solve systems of linear equations.