If we translate an application to a mathematical setup using two variables, then we need to form a linear system with two equations. Setting up word problems with two variables often simplifies the entire process, particularly when the relationships between the variables are not so clear.
The sum of \(4\) times a larger integer and \(5\) times a smaller integer is \(7\). When twice the smaller integer is subtracted from \(3\) times the larger, the result is \(11\). Find the integers.
Solution
Begin by assigning variables to the larger and smaller integer.
Let \(x\) represent the larger integer.
Let \(y\) represent the smaller integer.
When using two variables, we need to set up two equations. The first sentence describes a sum and the second sentence describes a difference.
This leads to the following system:
Solve using the elimination method. To eliminate the variable \(y\) multiply the first equation by \(2\) and the second by \(5\).
Add the equations in the equivalent system and solve for \(x\).
Back substitute to find \(y\).
\(\begin 4 x + 5 y & = 7 \\ 4 ( \color \color <)>+ 5 y & = 7 \\ 12 + 5 y & = 7 \\ 5 y & = - 5 \\ y & = - 1 \end\)
Answer:
The largest integer is \(3\) and the smaller integer is \(-1\).
An integer is \(1\) less than twice that of another. If their sum is \(20\), find the integers.
Answer
The two integers are \(7\) and \(13\).
Next consider applications involving simple interest and money.
A total of \($12,800\) was invested in two accounts. Part was invested in a CD at a \(3 \frac\)% annual interest rate and part was invested in a money market fund at a \(4 \frac\)% annual interest rate. If the total simple interest for one year was \($465\), then how much was invested in each account?
Solution
Begin by identifying two variables.
Let \(x\) represent the amount invested at \(3 \frac\)% \(= 3.125\) % \(= 0.03125\).
Let \(y\) represent the amount invested at \(4 \frac\)% \(= 4.75\)% = \(0.0475\).
The total amount in both accounts can be expressed as
To set up a second equation, use the fact that the total interest was \($465\). Recall that the interest for one year is the interest rate times the principal \((I = prt = pr ⋅ 1 = p)\). Use this to add the interest in both accounts. Be sure r to use the decimal equivalents for the interest rates given as percentages.
These two equations together form the following linear system:
Eliminate \(x\) by multiplying the first equation by \(-0.03125\).
Next, add the resulting equations.
Back substitute to find \(x\).
\(\begin x + y & = 12,800 \\ x + 4000 & = 12,800 \\ x & = 8,800 \end\)
Answer:
\($4,000\) was invested at \(4 \frac\)% and \($8,800\) was invested at \(3 \frac\)%.
A jar consisting of only nickels and dimes contains \(58\) coins. If the total value is \($4.20\), how many of each coin is in the jar?
Solution
Let \(n\) represent the number of nickels in the jar.
Let \(d\) represent the number of dimes in the jar.
The total number of coins in the jar can be expressed using the following equation:
Next, use the value of each coin to determine the total value \($4.20\).
This leads us to the following linear system:
Here we will solve using the substitution method. In the first equation, we can solve for \(n\).
Substitute \(n = 58 − d\) into the second equation and solve for \(d\).
\(\begin 0.05 ( \color\color < )>+ 0.10 d & = 4.20 \\ 2.9 - 0.05 d + 0.10 d & = 4.20 \\ 2.9 + 0.05 d & = 4.20 \\ 0.05 d & = 1.3 \\ d & = 26 \end\)
Now back substitute to find the number of nickels.
\(\begin n & = 58 - d \\ & = 58 - 26 \\ & = 32 \end\)
Answer:
There are \(32\) nickels and \(26\) dimes in the jar.
Joey has a jar full of \(40\) coins consisting of only quarters and nickels. If the total value is \($5.00\), how many of each coin does Joey have?
Answer
Joey has \(15\) quarters and \(25\) nickels.
Mixture problems often include a percentage and some total amount. It is important to make a distinction between these two types of quantities. For example, if a problem states that a \(20\)-ounce container is filled with a \(2\)% saline (salt) solution, then this means that the container is filled with a mixture of salt and water as follows:
| Percentage | Amount | |
|---|---|---|
| Salt | \(2\) \(= 0.02\) | \(0.02(20\) ounces\() = 0.4\) ounces |
| Water | \(98\)% \(= 0.98\) | \(0.98(20\) ounces\() = 19.6\) ounces |
In other words, we multiply the percentage times the total to get the amount of each part of the mixture.
A \(1.8\)% saline solution is to be combined and mixed with a \(3.2\)% saline solution to produce \(35\) ounces of a \(2.2\)% saline solution. How much of each is needed?
Solution
Let \(x\) represent the amount of \(1.8\)% saline solution needed.
Let \(y\) represent the amount of \(3.2\)% saline solution needed.
The total amount of saline solution needed is \(35\) ounces. This leads to one equation,
The second equation adds up the amount of salt in the correct percentages. The amount of salt is obtained by multiplying the percentage times the amount, where the variables \(x\) and \(y\) represent the amounts of the solutions. The amount of salt in the end solution is \(2.2\)% of the \(35\) ounces, or \(.022(35)\).
The algebraic setup consists of both equations presented as a system:
Add the resulting equations together
\(\begin - 0.018 x - 0.018 y &= - 0.63 \\ \pm\:\: 0.018 x + 0.032 y &= 0.77 \\ \hline \\0.014y &=0.14\\y&=\frac\\y&=10 \end\)
Back substitute to find \(x\).
Answer:
We need \(25\) ounces of the \(1.8\)% saline solution and \(10\) ounces of the \(3.2\)% saline solution.
An \(80\)% antifreeze concentrate is to be mixed with water to produce a \(48\)-liter mixture containing \(25\)% antifreeze. How much water and antifreeze concentrate is needed?
Solution
Let \(x\) represent the amount of \(80\)% antifreeze concentrate needed.
Let \(y\) represent the amount of water needed.
The total amount of the mixture must be \(48\) liters.
The second equation adds up the amount of antifreeze from each solution in the correct percentages. The amount of antifreeze in the end result is \(25\)% of \(48\) liters, or \(0.25(48)\).
Now we can form a system of two linear equations and two variables as follows:
Use the second equation to find \(x\):
\(\begin 0.80 x & = 12 \\ x & = \frac < 12 > < 0.80 >\\ x & = 15 \end\)
Back substitute to find \(y\).
\(\begin x + y & = 48 \\ \color + y & = 48 \\ y & = 33 \end\)
Answer:
We need to mix \(33\) liters of water with \(15\) liters of antifreeze concentrate.
A chemist wishes to create \(100\) ml of a solution with \(12\)% acid content. He uses two types of stock solutions, one with \(30\)% acid content and another with \(10\)% acid content. How much of each does he need?
Answer
The chemist will need to mix \(10\) ml of the \(30\)% acid solution with \(90\) ml of the \(10\)% acid solution.
Recall that the distance traveled is equal to the average rate times the time traveled at that rate, \(D = r ⋅ t\). These uniform motion problems usually have a lot of data, so it helps to first organize that data in a chart and then set up a linear system. In this section, you are encouraged to use two variables.
An executive traveled a total of \(4\) hours and \(875\) miles by car and by plane. Driving to the airport by car, she averaged \(50\) miles per hour. In the air, the plane averaged \(320\) miles per hour. How long did it take her to drive to the airport?
Solution
We are asked to find the time it takes her to drive to the airport; this indicates that time is the unknown quantity.
Let \(x\) represent the time it took to drive to the airport. Let \(y\) represent the time spent in the air.
Fill in the chart with the given information.
Use the formula \(D = r \cdot t\) to fill in the unknown distances.
The distance column and the time column of the chart help us to set up the following linear system.
Now back substitute to find the time \(x\) it took to drive to the airport:
Answer:
It took her \(1 \frac\) hours to drive to the airport.
It is not always the case that time is the unknown quantity. Read the problem carefully and identify what you are asked to find; this defines your variables.
Flying with the wind, a light aircraft traveled \(240\) miles in \(2\) hours. The aircraft then turned against the wind and traveled another \(135\) miles in \(1 \frac\) hours. Find the speed of the airplane and the speed of the wind.
Solution
Begin by identifying variables.
Let \(x\) represent the speed of the airplane.
Let \(w\) represent the speed of the wind.
Use the following chart to organize the data:
With the wind, the airplane’s total speed is \(x + w\). Flying against the wind, the total speed is \(x − w\).
Use the rows of the chart along with the formula \(D = r ⋅ t\) to construct a linear system that models this problem. Take care to group the quantities that represent the rate in parentheses.
If we divide both sides of the first equation by \(2\) and both sides of the second equation by \(1.5\), then we obtain the following equivalent system:
Here \(w\) is lined up to eliminate.
Answer:
The speed of the airplane is \(105\) miles per hour and the speed of the wind is \(15\) miles per hour.
A boat traveled \(27\) miles downstream in \(2\) hours. On the return trip, which was against the current, the boat was only able to travel \(21\) miles in \(2\) hours. What were the speeds of the boat and of the current?
Answer
The speed of the boat was \(12\) miles per hour and the speed of the current was \(1.5\) miles per hour.
Set up a linear system and solve.
14. A partitioned rectangular pen is constructed with a total \(168\) feet of fencing (see illustration). If the perimeter measures \(138\) feet, then find the dimensions of the pen.
15. Find \(a\) and \(b\) such that the system \(\left\ < \begin < l > < a x + b y = 8 >\\ < b x + a y = 7 >\end \right.\) has solution \((2,1)\). (Hint: Substitute the given \(x\)- and \(y\)-values and solve the resulting linear system in terms of \(a\) and \(b\).)
16. Find \(a\) and \(b\) such that the system \(\left\ < \begin < l > < a x - b y = 11 >\\ < b x + a y = 13 >\end \right.\) has solution \((3, -1)\).
17. A line passes through two points \((5, −9)\) and \((−3, 7)\). Use these points and \(y = mx + b\) to construct a system of two linear equations in terms of \(m\) and \(b\) and solve it.
18. A line passes through two points \((2, 7)\) and \((\frac, −2)\). Use these points and \(y = mx + b\) to construct a system of two linear equations in terms of \(m\) and \(b\) and solve it.
19. A \($5,200\) principal is invested in two accounts, one earning \(3\)% interest and another earning \(6\)% interest. If the total interest for the year is \($210\), then how much is invested in each account?
20. Harry’s \($2,200\) savings is in two accounts. One account earns \(2\)% annual interest and the other earns \(4\)%. His total interest for the year is \($69\). How much does he have in each account?
21. Janine has two savings accounts totaling \($6,500\). One account earns \(2 \frac\)% annual interest and the other earns \(3 \frac\)%. If her total interest for the year is \($211\), then how much is in each account?
22. Margaret has her total savings of \($24,200\) in two different CD accounts. One CD earns \(4.6\)% interest and another earns \(3.4\)% interest. If her total interest for the year is \($1,007.60\), then how much does she have in each CD account?
23. Last year Mandy earned twice as much interest in her Money Market fund as she did in her regular savings account. The total interest from the two accounts was \($246\). How much interest did she earn in each account?
24. A small business invested \($120,000\) in two accounts. The account earning \(4\)% annual interest yielded twice as much interest as the account earning \(3\)% annual interest. How much was invested in each account?
25. Sally earns \($1,000\) per month plus a commission of \(2\)% of sales. Jane earns \($200\) per month plus \(6\)% of her sales. At what monthly sales figure will both Sally and Jane earn the same amount of pay?
26. The cost of producing specialty book shelves includes an initial set-up fee of \($1,200\) plus an additional \($20\) per unit produced. Each shelf can be sold for \($60\) per unit. Find the number of units that must be produced and sold where the costs equal the revenue generated.
27. Jim was able to purchase a pizza for \($12.35\) with quarters and dimes. If he uses \(71\) coins to buy the pizza, then how many of each did he have?
28. A cash register contains \($5\) bills and \($10\) bills with a total value of \($350\). If there are \(46\) bills total, then how many of each does the register contain?
29. Two families bought tickets for the home basketball game. One family ordered \(2\) adult tickets and \(4\) children’s tickets for a total of \($36.00\). Another family ordered \(3\) adult tickets and \(2\) children’s tickets for a total of \($32.00\). How much did each ticket cost?
30. Two friends found shirts and shorts on sale at a flea market. One bought \(4\) shirts and \(2\) shorts for a total of \($28.00\). The other bought \(3\) shirts and \(3\) shorts for a total of \($30.75\). How much was each shirt and each pair of shorts?
31. A community theater sold \(140\) tickets to the evening musical for a total of \($1,540\). Each adult ticket was sold for \($12\) and each child ticket was sold for \($8\). How many adult tickets were sold?
32. The campus bookstore sells graphing calculators for \($110\) and scientific calculators for \($16\). On the first day of classes \(50\) calculators were sold for a total of \($1,646\). How many of each were sold?
33. A jar consisting of only nickels and quarters contains \(70\) coins. If the total value is \($9.10\), how many of each coin are in the jar?
34. Jill has \($9.20\) worth of dimes and quarters. If there are \(68\) coins in total, how many of each does she have?
Answer
1. The integers are \(16\) and \(29\).
3. The integers are \(6\) and \(35\).
5. The integers are \(25\) and \(36\).
7. The integers are \(−3\) and \(7\).
9. The integers are \(−5\) and \(−2\).
11. Length: \(17\) meters; width: \(6\) meters
13. Width: \(22\) feet; length: \(70\) feet
19. \($3,400\) at \(3\)% and \($1,800\) at \(6\)%
21. \($2,200\) at \(2 \frac\)% and \($4,300\) at \(3 \frac\)%
23. Savings: \($82\); Money Market: \($164\).
27. \(35\) quarters and \(36\) dimes
29. Adults \($7.00\) each and children \($5.50\) each.
31. \(105\) adult tickets were sold.
33. The jar contains \(42\) nickels and \(28\) quarters.
Set up a linear system and solve.
1. \(7\) gallons of the \(9\)% acid solution and \(1\) gallon of the \(17\)% acid solution
3. \(3.5\) pounds of the \(10\)% cashew mix and \(0.5\) pounds of the \(26\)% cashew mix
5. \(6\) ounces of cleaning fluid concentrate
7. \(18\) ounces of fruit juice concentrate and \(42\) ounces of water
Set up a linear system and solve.
1. The first leg of the trip took \(6\) hours and the second leg took \(2\) hours.
3. It took her \(\frac\) hour to drive to the airport.
5. \(0.5\) miles per hour.
7. Airplane: \(135\) miles per hour; wind: \(15\) miles per hour
11. One train averaged \(44\) miles per hour and the other averaged \(56\) miles per hour.
1. Answer may vary
3. Answer may vary
This page titled 3.3: Applications of Linear Systems with Two Variables is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Anonymous via source content that was edited to the style and standards of the LibreTexts platform.
