Download Word Problems in 3 Variables

Word Problems
Step 1:
Understand the problem.
Sometimes the problem lies in understanding the problem. If you are unclear as to what
needs to be solved, then you are probably going to get the wrong results. In order to show an
understanding of the problem you of course need to read the problem carefully. Sounds
simple enough, but some people jump the gun and try to start solving the problem before they
have read the whole problem. Once the problem is read, you need to list out all the
components and data that are involved. This is where you will be assigning your variables.
In the problems on this page, we will be letting each unknown be a separate
variable. So, if you have two unknowns, you will have two variables, x and y. If you have
three unknowns, you will have three variables, x, y, andz.
Step 2:
Devise a plan (translate).
When you devise a plan (translate), you come up with a way to solve the problem. Setting
up an equation, drawing a diagram, and making a chart are all ways that you can go about
solving your problem. In this tutorial, we will be setting up equations for each problem. In
the problems on this page, we will be setting up systems of linear equations. The number
of equations need to match the number of unknowns. For example, if you have two
variables, then you will need two equations. If you have three variables, then you will need
three equations.
Step 3:
Carry out the plan (solve).
The next step, carry out the plan (solve), is big. This is where you solve the system of
equations you came up with in your devise a plan step. The equations in the systems in this
tutorial will all be linear equations.
Step 4:
Look back (check and interpret).
You may be familiar with the expression don't look back. In problem solving it is good
to look back (check and interpret).. Basically, check to see if you used all your information
and that the answer makes sense. If your answer does check out make sure that you write
your final answer with the correct labeling.
Example 1: The sum of three numbers is 14. The largest is 4 times the smallest, while the
sum of the smallest and twice the largest is 18. Find the numbers.
Step 1:
Understand the problem.
Make sure that you read the question carefully several times.
Since we are looking for three numbers, we will let
x = the smallest number
y = middle number
z = the largest number
Step 2:
Devise a plan (translate).
Since we have three unknowns, we need to build a system with three equations.
Equation (1):
Equation (2):
Equation (3):
Putting the three equations together in a system we get:
Step 3:
Carry out the plan (solve).
This is a system of linear equations with three variables, which can be found in Tutorial
20: Solving Systems of Linear Equations in Three Variables.
Simplify and put all three equations in the form Ax + By + Cz = D if needed.
Equation (2) needs to be put in the correct form:
(1)
(2)
(3)
*subtract 4x from both sides so it is in standard
form (Ax + By + Cz = D)
Equation (4) is just Equation (2) in the correct form.
Choose to eliminate any one of the variables from any pair of equations.
Since y is already eliminated in equation (4) and (3), it would be quickest and easiest to
eliminate y.
We can use equation (4) as one equation with y eliminated:
*y is already eliminated
Eliminate the SAME variable chosen in step 2 from any other pair of equations, creating
a system of two equations and 2 unknowns.
We can use equation (3) as another equation with y eliminated:
*y is already eliminated
Solve the remaining system found in step 2 and 3.
Putting those two equations together we get:
I’m going to choose to eliminate z.
Multiplying equation (4) by -2 and then adding that to equation (3) we get:
*Mult. both sides of eq. (4) by -2
*z's have opposite coefficients
*z's dropped out
Solving for x we get:
* divide both sides by 9
Using equation (4) to plug 2 in for x and solving for z we get:
*Equation (4)
*Plug in 2 for x
*add 8 to both sides
Solve for the third variable.
Using equation (1) to plug in 2 for x and 8 for z and solving for y we get:
*Equation (1)
*Plug in 2 for x and 8 for z
*subtract 10 from both sides
Step 4:
Look back (check and interpret).
You will find that if you plug the ordered triple (2, 4, 8) into ALL THREE equations of the
original system, this is a solution to ALL THREE of them.
Final Answer:
2 is the smallest number, 4 is the middle number and 8 is the largest number.
Example 2: A landlord owns 3 condominiums, a 1-bedroom condo, a 2-bedroom condo,
and a 3-bedroom condo. The total rent she receives is $1240. She needs to make repairs on the condos,
and it costs 10% of the 1-bedroom condo’s rent for its repairs, 20% of the 2-bedroom for its repairs, and
30% of the 3-bedroom condo’s rent for its repairs. The total repair bill was $276. The 3-bedroom condo’s
rent is twice the 1-bedroom condo’s rent. How much is the rent for all three condos?
Step 1:
Understand the problem.
Make sure that you read the question carefully several times.
Since we are looking for three different rents, we will let
x = the 1-bedroom condo’s rent
y = the 2-bedroom condo’s rent
z = the 3 bedroom condo’s rent
Step 2:
Devise a plan (translate).
Since we have three unknowns, we need to build a system with three equations.
Equation (1):
The total rent is $1240
x + y + z = 1240
Equation (2):
10% of x + 20% of y + 30% of z is $276
.1x + .2y + .3z = 278
Equation (3):
The 3-bedroom condo’s rent is twice the 1-bedroom condo’s rent
z = 2x
Putting the three equations together in a system we get:
x + y + z = 1240
Eq (1)
.1x + .2y + .3z = 276
Eq (2)
z = 2x
Eq (3)
Step 3:
Carry out the plan (solve).
This is a system of linear equations with three variables, which can be found in Tutorial
20: Solving Systems of Linear Equations in Three Variables.
Since the last equation has only 2 variables, it is easy to put one in terms of the other (it’s
already in that format) and substitute that value into the other 2 equations.
z=2x
Substitute 2x for z in the other two equations.
(1) x + y + 2x = 1240
(2) .1x + .2y + .3(2x) = 276
3x + y = 1240
.7x + .2y =278
10(.7x + .2y) =276(10)
3x + y = 1240
(Eq 4)
7x + 2y = 2760 (Eq 5)
Combine like terms and simplify if possible
To get rid of decimals, multiply both sides of the 2nd
equation by 10.
Equation 4 is just Equation1 with 2x substituted for z
Equation 5 is just Equation 2 with 2x substituted for z
Choose to eliminate any one of the variables from any pair of equations.
Since z is already eliminated in equation (4) and (5), it would be quickest and easiest to
eliminate y. Coefficients of y are 1y and 2y. LCM = 2. Goal: Get coefficients to be 2 and -2
-2(3x + y) = 1240 (-2)
7x + 2y = 2760
-6x - 2y = -2480
x
= $280
After multiplying Eq (4) by -2, add it to Eq (5) to eliminate y
Using equation (4) or Equation (5) to plug 280 in for x and solving for y we get:
3(280) + y = 1240
(Eq 4)
*Equation (4)
*Plug in 280 for x
*subtract 280 from both sides
840 + y = 1240
y = $400
Solve for the third variable.
Using equation (3) to plug in 280 for x and solve for z we get:
*Equation (1)
*Plug in 280 for x and solve for z
z = 2x
z = 2(280)
z = $560
Step 4:
*
Look back (check and interpret).
You will find that if you plug the ordered triple (280, 400, 520) into ALL THREE equations
of the original system, this is a solution to ALL THREE of them.
280 + 400 + 560 = 1240
Yes
.1(280) + .2(400) + .3(560) = 276
Yes
560 = 2(280)
Yes
Final Answer:
$280 is the 1-bedroom condo’s rent
$400 is the 2-bedroom condo’s rent
$560 is the 3 bedroom condo’s rent