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 does 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?

### Adding or Subtracting Linear Systems

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.

#### Let's solve the following problems using systems of equations and the Elimination Method:

- 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?

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*}

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

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).

- 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.

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*}.

### Examples

#### Example 1

Earlier, you were told that a family going to an amusement park bought 5 adult tickets and 8 children's tickets and paid $124 while another family bought 5 adult tickets and 12 children's tickets and paid $156. How much does each type of ticket cost?

Let \begin{align*}a\end{align*} represent the cost of adult tickets and \begin{align*}c\end{align*} represent the cost of children's tickets. The system of equations that represents this situation is:

\begin{align*}\begin{cases} 5a+8c=124\\ 5a+12c=156\end{cases}\end{align*}

Since these equations are in standard form and both have the term \begin{align*}5a\end{align*}, we can use the Elimination Method and subtract the equations:

\begin{align*}& \qquad \ 5a+8c=124\\ &\underline{\;\; - \ (5a+12c = 156) \;\;}\\ & \qquad \ 0a-4c =-32\\ & \qquad \ -4c=-32\\ & \qquad \ c=8\end{align*}

Now, substitute the value for \begin{align*}c\end{align*} in to solve for \begin{align*}a\end{align*}:

\begin{align*}5a + 8(8) &= 124\\ 5a + 64 &=124\\ 5a &=60\\ a &=12\end{align*}

The adult tickets cost $12 and the children's tickets cost $8.

#### Example 2

Solve the system \begin{align*}\begin{cases} 5s+2t=6\\ 9s+2t=22\end{cases}\end{align*}.

Since these equations are both written in standard form, and both have the term \begin{align*}2t\end{align*} in them, we will will use elimination by subtracting. This will cause the \begin{align*}t\end{align*} terms to cancel out and we will be left with one variable, \begin{align*}s\end{align*}, which we can then isolate.

\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*}.

### Review

- 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\\ 8x-3y=-19 \end{cases}\end{align*}
- \begin{align*}\begin{cases} x+4y=-9\\ -2x-5y=12 \end{cases}\end{align*}
- \begin{align*}\begin{cases} -2x-5y=-10\\ x+4y=8 \end{cases}\end{align*}
- \begin{align*}\begin{cases} x-3y=-10\\ -8x+5y=-15 \end{cases}\end{align*}
- \begin{align*}\begin{cases} -x-6y=-18\\ x-6y=-6 \end{cases}\end{align*}
- \begin{align*}\begin{cases} 5x-3y=-14\\ x-3y=2 \end{cases}\end{align*}
- \begin{align*}&3x+4y=2.5\\ &5x-4y=25.5 \end{align*}
- \begin{align*}&5x+7y=-31\\ &5x-9y=17 \end{align*}
- \begin{align*}&3y-4x=-33\\ &5x-3y=40.5 \end{align*}
- 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?
- 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.
- 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:
- the pick-up fee
- the per-mile rate
- the cost of a seven-mile trip

- Calls from a call-box are charged per minute at one rate for the first five minutes, and then at 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.
- 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 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**

- 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 8-foot 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 5.5-week mark?
- Is there a restriction on how high the plant will grow? Does your equation show this?

- The “Wave” is an exciting pastime 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*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 7.4.