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7.3: Solving Linear Systems by Addition or Subtraction

Difficulty Level: At Grade Created by: CK-12
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As you noticed in the previous lesson, 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 1: 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*} the number of apples and \begin{align*}b=\end{align*} the number of bananas. By translating each purchase into an equation, you get the following system:

\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*}. If you were to subtract these equations, what would happen?

\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: Solve the system \begin{align*}\begin{cases} 3x+2y=11\\ 5x-2y=13\end{cases}\end{align*}.

Solution: These equations would take much more work to rewrite in slope-intercept 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*}variables have nothing in common, so adding will not cancel the \begin{align*}x-\end{align*}variable.

Looking at the \begin{align*}y-\end{align*}variable, you can see the coefficients are 2 and –2. By adding these together, you get zero. Add these two equations and see what happens.

\begin{align*}& \qquad \ 3x+2y =11\\ &\underline{\;\; + \ (5x-2y) =13 \;\;}\\ & \qquad \ 8x+0y =24\end{align*}

The resulting equation is \begin{align*}8x=24\end{align*}. Solving for \begin{align*}x\end{align*}, you get \begin{align*}x=3\end{align*}. To find the \begin{align*}y-\end{align*}coordinate, choose either equation, and substitute the number 3 for the variable \begin{align*}x\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: Andrew is paddling his canoe down a fast-moving 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*}, and the speed of the river \begin{align*}y\end{align*}. When traveling downstream, Andrew's speed is boosted by the river current, so his total speed is the canoe speed plus the speed of the river \begin{align*}(x+y)\end{align*}. Upstream, his speed is hindered by the speed of the river. His speed upstream is \begin{align*}(x-y)\end{align*}.

\begin{align*}\text{Downstream Equation} && x+y&=7\\ \text{Upstream Equation} && x-y&=1.5\end{align*}

Notice \begin{align*}y\end{align*} and \begin{align*}-y\end{align*} are additive inverses. If you add them together, their sum equals zero. Therefore, by adding the two equations together, the variable \begin{align*}y\end{align*} will cancel, leaving you to solve for the remaining variable, \begin{align*}x\end{align*}.

\begin{align*}& \qquad \ x+y=7\\ &\underline{\;\; + \ (x-y)=1.5 \;\;}\\ & \quad \ 2x+0y=8.5\\ & \qquad \quad \ \ 2x=8.5\end{align*}

Therefore, \begin{align*}x=4.25\end{align*}; Andrew is paddling \begin{align*}4.25 \ miles/hour\end{align*}. To find the speed of the river, substitute your known value into either equation and solve.

\begin{align*}4.25-y&=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*}.

Multimedia Link: For more 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.

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 Elimination (12:44)

  1. 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.

  1. \begin{align*}\begin{cases} 2x+y=-17\\ 8x-3y=-19 \end{cases}\end{align*}
  2. \begin{align*}\begin{cases} x+4y=-9\\ -2x-5y=12 \end{cases}\end{align*}
  3. \begin{align*}\begin{cases} -2x-5y=-10\\ x+4y=8 \end{cases}\end{align*}
  4. \begin{align*}\begin{cases} x-3y=-10\\ -8x+5y=-15 \end{cases}\end{align*}
  5. \begin{align*}\begin{cases} -x-6y=-18\\ x-6y=-6 \end{cases}\end{align*}
  6. \begin{align*}\begin{cases} 5x-3y=-14\\ x-3y=2 \end{cases}\end{align*}
  7. \begin{align*}3x+4y=2.5\!\\ 5x-4y=25.5 \end{align*}
  8. \begin{align*}5x+7y=-31\!\\ 5x-9y=17 \end{align*}
  9. \begin{align*}3y-4x=-33\!\\ 5x-3y=40.5 \end{align*}
  10. Nadia and Peter visit the candy store. Nadia buys three candy bars and four fruit roll-ups for $2.84. Peter also buys three candy bars, but he can afford only one fruit roll-up. His purchase costs $1.79. What is the cost of each candy bar and each fruit roll-up?
  11. 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 air-traffic controller reads its ground-speed (speed measured relative to the ground) at 275 miles per hour. Another identical plane moving in the opposite direction has a ground-speed of 227 miles per hour. Assuming both planes are flying with identical air-speeds, calculate the speed of the wind.
  12. An airport taxi firm charges a pick-up fee, plus an additional per-mile fee for any rides taken. If a 12-mile journey costs $14.29 and a 17-mile journey costs $19.91, calculate:
    1. the pick-up fee
    2. the per-mile rate
    3. the cost of a seven-mile trip
  13. Calls from a call-box are charged per minute at one rate for the first five minutes, then a different rate for each additional minute. If a seven-minute call costs $4.25 and a 12-minute call costs $5.50, find each rate.
  14. 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?
  15. Paul has a part-time 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

  1. Graph \begin{align*}y=|x|-5\end{align*}.
  2. Solve. Write the solution using interval notation and graph the solution on a number line: \begin{align*}-9\ge \frac{c}{-4}\end{align*}.
  3. The area of a rectangle is 1,440 square centimeters. Its length is ten times more than its width. What are the dimensions of the rectangle?
  4. Suppose \begin{align*}f(x)=8x^2-10\end{align*}. Find \begin{align*}f(-6)\end{align*}.
  5. Torrey is making candles from beeswax. Each taper candle needs 86 square inches and each pillar candle needs 264 square inches. Torrey has a total of 16 square feet of beeswax. Graph all the possible combinations of taper and pillar candles Torrey could make (Hint: \begin{align*}one \ square \ foot=144 \ square \ inches\end{align*}).

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