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, and z.

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.

Since we are looking for three numbers, we will let 

x = the smallest number

y = middle number

z = the largest number

Equation (1):

Equation (2):

Equation (3):

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:

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:

We can use equation (3) as another equation with y eliminated:

Putting those two equations together we get:

Multiplying equation (4) by -2 and then adding that to equation (3) we get:

*z's have opposite coefficients

*z's dropped out

*Inverse of sub. 8 is add 8
 

Using equation (1) to plug in 2 for x and 8 for z and solving for y we get:

*Inverse of add 10 is sub. 10
 

Final Answer:
2 is the smallest number, 4 is the middle number and 8 is the largest number.

Since we are looking for two different amounts, we will let 

x = the number of gallons of 20% alcohol solution

y = the number of gallons of 50% alcohol solution

Equation (1):

Equation (2):

Simplify if needed. 

We can simplify this by multiplying both sides of equation (2) by 10 and getting rid of the decimals:

Multiply one or both equations by a number that will create opposite coefficients for either x or y if needed AND add the equations.

If we multiply equation (1) by -2, then the x’s will have opposite coefficients. 

Multiplying equations (1) by -2 and then adding that to equation (3) we get:

*x's have opposite coefficients
 

*x's dropped out

Solving for y we get:

Using equation (1) to plug in 3 for y and solving for x we get:

*Inverse of add 3 is sub. 3
 

Final Answer:
6 gallons of 20% solution and 3 gallons of 50% solution

Since we are looking for two different rates, we will let 

x = rate of the plane

y = the rate of the wind

Since this is a rate/distance problem, it might be good to organize our information using the distance formula.

Keep in mind that the wind speed is affecting the overall speed. 

When the plane is with the wind, it will be going faster.  That rate will be x + y. 

When the plane is going against the wind, it will be going slower.  That rate will be x - y.
 
 

Equation (1):

Equation (2):

Simplify if needed. 

We can simplify this by dividing both sides of equation (1) by 2 and equation (2) by 2.5 getting rid of the ( ) and decimals at the same time:

*Div. both sides of eq. (2) by 2.5
 
 
 

Multiply one or both equations by a number that will create opposite coefficients for either x or y if needed AND add the equations.

Since we already have opposite coefficients on y, we can go right into adding equations (3) and (4) together:

*y's dropped out

Solving for x we get:

Using equation (3) to plug in 270 for x and solving for y we get:

Final Answer:
The airplane speed is 270 mph and the air speed is 30 mph

Example 4:    Given the cost function C(x) and the revenue function R(x), find the number of units x that must be sold to break even.

C(x) = 20x + 50000
R(x) = 25x

We will let,

x = the number of units

C(x) = 20x + 50000

R(x) = 25x

This problem appears a little different because of the function notation. Keep in mind that function notation translates to being y.

Cost function:
C(x) = 20x + 50000

Revenue function:
R(x) = 25x

Since the break-even point is when revenue = cost, we will go right into setting this up using the substitution method as discussed in Tutorial 19: Solving Systems of Linear Equations in Two Variables.

*Inverse of mult. by 5 is div. by 5

Final Answer:
10000 units are needed to break-even

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem.  At the link you will find the answer as well as any steps that went into finding that answer.