Download SOLVING SYSTEMS OF EQUATIONS BIG Picture: Solving a system

SOLVING SYSTEMS OF EQUATIONS
BIG Picture: Solving a system of equations means finding the values of the variables that are
true in both equations.
Both the Substitution Method and Elimination Method involve removing one variable so that
we can solve for the other variable.
The Substitution Method
1. If there are fractions in the equations, clear the fractions.
2. Solve one of the equations for one of the variables.
It is easiest to solve for a variable that has a coefficient of ±1, if possible.
3. Substitute the equivalent expression for the variable found in Step 2 into the other
equation.
There should be only one variable now.
4. Solve the equation from Step 3 for the remaining variable.
5. Substitute the value of the variable from Step 4 into the equation from Step 2 and
solve. This will give you the value of the other variable.
6. The solution is the ordered pair (x, y).
Check the solution in both of the original equations.
Example:
Solve the system of equations by substitution:
2x - y = 9
Eq. 1
3x + 2y = 10
Eq. 2
I choose to solve for y in Eq. 1 because it has a coefficient of -1. This will prevent me
from having to deal with fractions, at least initially. I may encounter them later.
2x - y = 9
Eq. 1
-y = -2x + 9
Subtracting 2x from both sides.
y = 2x - 9
Eq. 3
Dividing (or multiplying) both sides by -1.
3x + 2(2x - 9) = 10
3x + 4x - 18 = 10
7x - 18 = 10
7x = 28
x=4
y = 2x - 9
= 2(4) - 9
=8-9
= -1
Plugging in equivalent expression, 2x - 9, in place
of y in Eq. 2.
Distributing.
Combining like terms.
Adding 18 to both sides.
Dividing by 7 on both sides.
Returning to Eq. 3.
Substituting 4 for x.
The solution is (4, -1)
Check the solution by substituting x and y into both of the original equations.
Check the solution again by graphing both lines and finding the intersection point.
The Elimination Method
1. If there are fractions in the equations, clear the fractions.
2. Rewrite each equation in the form Ax + By = C, if necessary.
3. Select one variable for elimination, x or y.
4. Multiply one or both equations by appropriate numbers so that the coefficients of
the selected variable are opposites of one another.
5. Add the two equations. This will eliminate the chosen variable.
6. Solve the resulting equation for the remaining variable.
7. Substitute this value into one of the original equations and solve for the other
variable.
8. The solution is the ordered pair (x, y).
Check the solution in both of the original equations.
Example:
Solve the system of equations by elimination:
6x - 5y = 12
Eq. 1
7x - 4y = 3
Eq. 2
I choose to eliminate the y variables so I need to find a number that is divisible by both
-5 and -4 (the coefficients of the y variables). I select 20.
Multiplying Eq. 1 by 4 will make the coefficient of that y variable -20
Multiplying Eq. 2 by -5 will make the coefficient of the y variable 20.
This is the condition I need to eliminate the y's.
6x - 5y = 12
7x - 4y = 3
(4)
(-5)
24x - 20y = 48
-35x + 20y = -15
-11x
= 33
x
= -3
Adding equations.
Dividing both sides by -11.
To find y, substitute the value of x into one of the original equations. I select Eq. 1.
6(-3) - 5y = 12
-18 - 5y = 12
-5y = 30
y = -6
Adding18 to both sides.
Dividing both sides by -5.
The solution is (-3, -6)
Check the solution by substituting x and y into both of the original equations.
Check the solution again by graphing both lines and finding the intersection point. (You
will need to convert the equations to y = mx + b form first.)