Systems of Equations

Graphing systems of equations can't always get the exact answer. The method of graphing the systems of equations usually leads to estimating the answer.  Substitution allows us to algebraically find the exact answer.  For substitution, one variable in one equation needs to be isolated, and the substituted into the other equation.  Some methods to help isolating the variable is to isolate the variable with the coefficient of one.  For example, 3x + y = 15, isolate the y.  

 

3x + y = 15

3x - 3x + y = 15 - 3x

y = -3x + 15

 

Once that variable is isolated, substitute it in the other equation and solve for x.

 

5x + y = 25

5x + (-3x + 15) = 25

2x + 15 = 25

2x + 15 - 15 = 25 - 15

2x = 10

2x/2 = 10/2

x = 5

 

After it is substituted in, find the value to y.  

 

y = -3x + 15

y = -3(5) + 15

y = -15 + 15

y = 0

 

Another method for substitution is when the coefficient is greater than one, choose the coefficient that is easier to isolate. It is easier to isolate when the numbers are divisble.

 

2x + 4y = 8

3x + 2y = 7

 

2x + 4y - 4y = 8 - 4y

2x/2 = 8 - 4y/2

x = 4 - 2y

 

3x + 2y = 7

3(4 - 2y) + 2y = 7

12 - 6y + 2y = 7

12 - 4y = 7

12 - 12 - 4y = 7 - 12

-4y = -5

-4y/-4 = -5/-4

y = 5/4

 

x = 4 - 2(5/4)

x = 4 - 5/2

x = 3/2

 

Sometimes it won't always come out to exact solutions. When the equations are the same, or when the statement is true, then there are infinite solutions.  If the statment is false, then there are no solutions.

 

No solution example:

 

3x + 2 = 15

3x - 3 = 15

 

This systems of equations has the same slope,

and therefore have no solution.