Solving Systems Using Substitution

Due to math content, this page has special requirements (including JavaScript) for full functionality.
With your current viewing scenario, it is not appearing and behaving as it is supposed to!
Please visit Dr. Carol J.V. Fisher's Homepage to learn what this site has to offer.
Watch the "Welcome" video to get started—hope to see you back here soon!

Dr. Carol J.V. Fisher's Homepage

For this exercise, you need ♥ INTERNET EXPLORER 6.0 and above, with MathPlayer installed.♥

SOLVING SYSTEMS USING SUBSTITUTION

Jump right to the exercises!

One common method for solving systems is called substitution,
because information from one equation is substituted into other equations.

The technique of substitution can be used for a wide variety of systems:
any number of equations or inequalities, any number of unknowns, linear or nonlinear.
However, in this exercise, the technique is only applied to systems of two linear equations in two unknowns.

The technique of substitution works best when one of the equations can easily be solved for one of the variables.
For example, maybe the first equation can easily be solved for x .
Or, perhaps the second equation can easily be solved for y .

Remember: to solve for a variable means to get it all by itself, on one side of the equation.
The variable you're solving for is not allowed to appear on the other side.

Here are some systems which are ideally suited to the substitution technique:

y=3x- 1
5x+7y =4
(First equation is already solved for y .)

3y+2x =8-4y
x-7y=6
(Second equation can easily be solved for x .)

Here are the general steps for solving a system of two equations in two unknowns using substitution:

Solve one of the equations for one of the unknowns.
Pick the simplest equation to work with!
Substitute this information into the other equation.
At this point, you will have an equation with only one variable.
Solve for this variable.
(Note: If the equation is always false, then there are no solutions. The lines are parallel.
If the equation is always true, then there are infinitely many solutions. The lines are the same.)
Go back and substitute this information into the original equation.
Clearly report your solution(s).
Be sure to give exact answers, not decimal approximations.
Of course, you may use your calculator to check your results.

EXAMPLE (a system with a unique solution):
Solve the system:
y=3x- 1
5x+7y =4

5x+7( 3x-1)= 4	The first equation is already solved for y . Substitute it into the second equation.
5x+21x -7=4 26x=11 x=11 26	Solve the equation for x .
y=3( 1126) -1=33 26-26 26=7 26	Substitute this value of x back into the original equation. Simplify.
The unique solution is the ordered pair (11 26,7 26) .	Clearly report the solution.

EXAMPLE (a system with no solutions):
Solve the system.
Report your solution(s) in the form (u,q) .
q+7u=- 4
-4q+20 =28u

q= -4-7u	Solve the first equation for q .
-4( -4-7u ) +20 =28u	Substitute this expression for q into the second equation.
16+28u +20=28u 36=0	Simplify. The resulting equation is always false. There are no solutions. These two lines are parallel.

EXAMPLE (a system with infinitely many solutions):
Solve the system.
Report your solution(s) in the form (s,t) .
2t=8s+ 12
t-9=4 s-3

t= 4s+6	Solve the second equation for t .
2( 4s+6 ) =8s+12	Substitute this expression for t into the first equation.
8s+12 =8s+12 0=0	Simplify. The resulting equation is always true. These two lines are the same. There are infinitely many solutions.
All points of the form (s, 4s+6) are solutions.	Report the entire line of solutions.

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.