Applications of linear systems in three unknowns
Interactive practice questions
A theatre can seat a total of $705$705 people. Tickets cost $\$2$$2 for children, $\$4$$4 for students, and $\$6$$6 for adults. At a recent sold out screening, the combined total of children and students was equal to twice the number of adults. The money made from ticket sales totalled $\$2842$$2842.
Let $x$x be the number of children, $y$y be the number of students, and $z$z be the number of adults.
Complete the system of equations:
| $x$x | $+$+ | $y$y | $+$+ | $z$z | $=$= | $\editable{}$ |
| $\editable{}x$x | $+$+ | $\editable{}y$y | $+$+ | $\editable{}z$z | $=$= | $\editable{}$ |
| $x$x | $+$+ | $y$y | $=$= | $\editable{}z$z |
In the following subproblems we will complete one approach to solving the system of equations using the elimination method:
| $x$x | $+$+ | $y$y | $+$+ | $z$z | $=$= | $705$705 | ----- equation $1$1 |
| $2x$2x | $+$+ | $4y$4y | $+$+ | $6z$6z | $=$= | $2842$2842 | ----- equation $2$2 |
| $x$x | $+$+ | $y$y | $=$= | $2z$2z | ----- equation $3$3 |
Notice that $x+y$x+y appears in equation $1$1 and in equation $3$3. In equation $3$3, $x+y=2z$x+y=2z, and so we can make a substitution in place of $x+y$x+y in equation $1$1.
Step 1: Substitute equation $3$3 into equation $1$1.
| $2$2$\editable{}$ | $+$+ | $z$z | $=$= | $705$705 |
| $\editable{}$ | $=$= | $705$705 | ||
| $z$z | $=$= | $\editable{}$ |
Now that we have a value for $z$z, we can plug this value into equation $1$1.
Step 2: Substitute the value of $z$z into equation $1$1.
| $x$x | $+$+ | $y$y | $+$+ | $\editable{}$ | $=$= | $705$705 | ----- equation $1$1 |
| $x$x | $+$+ | $y$y | $=$= | $\editable{}$ | ----- equation $4$4 |
We can also plug this value of $z$z into equation $2$2.
Step 3: Substitute the value of $z$z into equation $2$2.
| $2x$2x | $+$+ | $4y$4y | $+$+ | $6\times\editable{}$6× | $=$= | $2842$2842 | ---- equation $2$2 |
| $2x$2x | $+$+ | $4y$4y | $+$+ | $\editable{}$ | $=$= | $2842$2842 | |
| $2x$2x | $+$+ | $4y$4y | $=$= | $\editable{}$ | ----- equation $5$5 |
Now we have two linear equations in two unknowns ($x$x and $y$y).
We would like to use the elimination method here, so we need both of the equations to have the same coefficient for one of the unknowns. Let's make the $x$x coefficient the same by multiplying every term in equation $4$4 by $2$2.
Step 4: Multiply equation $4$4 by $2$2 so we can use the elimination method.
| $x$x | $+$+ | $y$y | $=$= | $470$470 | ----- equation $4$4 |
| $2x$2x | $+$+ | $2y$2y | $=$= | $\editable{}$ | ----- equation $6$6 |
Now that equation $4$4 and equation $6$6 have the same $x$x coefficient we can subtract one from the other to get an equation with only one unknown.
Step 5: Subtract equation $6$6 from equation $5$5.
| $2x$2x | $+$+ | $4y$4y | $=$= | $1432$1432 | ----- equation $5$5 |
| $2x$2x | $+$+ | $2y$2y | $=$= | $940$940 | ----- equation $6$6 |
| $\editable{}y$y | $=$= | $\editable{}$ | |||
| $y$y | $=$= | $\editable{}$ |
Now we have a value for $y$y, we can substitute this value back into equation $4$4.
Step 6: Substitute the value of $y$y into equation $4$4.
| $x$x | $+$+ | $\editable{}$ | $=$= | $470$470 | ----- equation $4$4 |
| $x$x | $=$= | $\editable{}$ |
Deduce the number of:
Children = $\editable{}$.
Students = $\editable{}$.
Adults = $\editable{}$.
Hartman Rent-A-Car are about to spend $\$2008000$$2008000 buying a fleet of new vehicles. The fleet is made up of hatchbacks, sedans and SUVs. Hatchbacks cost $\$10000$$10000 each, sedans cost $\$13000$$13000 each and SUVs cost $\$17000$$17000 each.
Hartman Rent-A-Car will purchase twice as many hatchbacks as sedans and the total number of cars to be bought is $152$152.
For the opening night of Serse' at the Opera House, a total of $582$582 tickets were sold.
Box seat tickets cost $\$52$$52, stall seat tickets cost $\$72$$72, and balcony seat tickets cost $\$80$$80. There were $446$446 more box and stall seat tickets sold than there were balcony seat tickets sold.
The total money made from ticket sales for the night was $\$36548$$36548.
Paul has made three different investments. In the last financial year he earned $7%$7% interest on his savings account, $1%$1% interest on his mutual funds investment, and $2%$2% interest on his money market investment. Paul earned a total of $\$45600$$45600 in interest from the investments.
Paul has twice as much money invested in the money market than he has in his savings account. Further, he earned half as much from his money market investment than he did from his mutual funds investment.