Download solving systems of equations

East Campus, CB 117
361-698-1579
Math Learning Center
SOLVING SYSTEMS OF EQUATIONS
West Campus, HS1 203
361-698-1860
This handout covers four methods to solve a system of equations:
1) Graphing
2) Substitution
3) Elimination or addition method
4) TI-83/TI-84 Matrix Solution
For Example:
6𝑥𝑥 + 3𝑦𝑦 = 12
5𝑥𝑥 + 𝑦𝑦 = 7
Method 1 – Graphing
To solve by graphing, the two lines must be graphed on the same rectangular coordinate system. To graph each
line, you need to find at least two points on each line. Set up and x & y chart for each. The point at which the
two graphs intersect is the solution.
(select a value for one variable and solve for the other,
Remember: these are just my choices)
For
6𝑥𝑥 + 3𝑦𝑦 = 12
x y
0 4
2 0
For
x
0
1.4
5𝑥𝑥 + 𝑦𝑦 = 7
y
7
0
Method 2 – Substitution
To solve by substitution, select one equation (it does not matter which equation) and solve for one of the
variables (it does not matter which variable). Then substitute that expression in to the other equation.
For 6𝑥𝑥 + 3𝑦𝑦 = 12, Let’s solve for y.
6𝑥𝑥 + 3𝑦𝑦 = 12
3𝑦𝑦 = −6𝑥𝑥 + 12
−6𝑥𝑥
− 6𝑥𝑥
3𝑦𝑦
3
=
−6𝑥𝑥+12
𝑦𝑦 =
3
−6𝑥𝑥
3
+
12
3
𝑦𝑦 = −2𝑥𝑥 + 4
Since I solved for y, I’m going to use this to substitute in to the y variable of the second equation.
For
5𝑥𝑥 + 𝑦𝑦 = 7
5𝑥𝑥 + (−2𝑥𝑥 + 4) = 7
5𝑥𝑥 + -2𝑥𝑥 + 4 = 7
Now solve for x.
3𝑥𝑥 + 4 = 7
3𝑥𝑥 = 3
−4 − 4
3𝑥𝑥
3
3
=3
𝑥𝑥 = 1
Now that we know what x is, use this to solve for y by using one of the original equations (it does not matter
which equation). Let’s use 6𝑥𝑥 + 3𝑦𝑦 = 12.
6(1) + 3𝑦𝑦 = 12
3𝑦𝑦
6
6 + 3𝑦𝑦 = 12
3𝑦𝑦 = 6
=3
𝑦𝑦 = 2
3
−6
−6
Therefore, our answer is (1,2) which happens to be the point of intersection.
East Campus, CB 117
361-698-1579
Math Learning Center
West Campus, HS1 203
361-698-1860
Method 3: Elimination or Addition Method
The object of the elimination method is to add the equations such that like terms add up to zero. If no like terms
add up to zero, then multiplication is used to alter the system of equations such that like terms do add up to
zero.
If the equations 6𝑥𝑥 + 3𝑦𝑦 = 12 were added now, no like terms would add up to zero.
+ 5𝑥𝑥 + 𝑦𝑦 = 7
11𝑥𝑥 + 4𝑦𝑦 = 19
Therefore, multiplication is needed to alter the equations. To do this legally, remember that whatever you do to
one side of an equation you do the same thing on the other side of the equation. To eliminate a variable do the
following:
1) Decide which variable you want to eliminate (it does not matter which variable)
6𝑥𝑥 + 3𝑦𝑦 = 12
5𝑥𝑥 + 𝑦𝑦 = 7
Let’s pick the x terms.
2) Looking at the variable from each equation, find the Least Common Multiple of the coefficients
𝟔𝟔𝑥𝑥 + 3𝑦𝑦 = 12
𝟓𝟓𝑥𝑥 + 𝑦𝑦 = 7
The Least Common Multiple of 6 and 5 is 30.
3) Now multiply each equation by the number which will make the coefficients on the x variables 30
𝟓𝟓(𝟔𝟔𝑥𝑥 + 3𝑦𝑦 = 12)
30𝑥𝑥 + 15𝑦𝑦 = 60
−𝟔𝟔(𝟓𝟓𝑥𝑥 + 𝑦𝑦 = 7)
+ − 30𝑥𝑥 − 6𝑦𝑦 = −42
Notice that if these equations were added, the x terms would still not add to zero, therefore one of the factors,
5 or 6, which was used to multiply with the equations must be negative. Let’s choose the 6 to be negative.
𝟓𝟓(𝟔𝟔𝑥𝑥 + 3𝑦𝑦 = 12)
−𝟔𝟔(𝟓𝟓𝑥𝑥 + 𝑦𝑦 = 7)
30𝑥𝑥 + 15𝑦𝑦 = 60
+ − 30𝑥𝑥 − 6𝑦𝑦 = −42
9𝑦𝑦 = 18
9𝑦𝑦
9
=
18
9
𝑦𝑦 = 2
4) Now substitute this value into one of the original equations (it does not matter which one) to solve for x. Let’s
use 6𝑥𝑥 + 3𝑦𝑦 = 12.
6𝑥𝑥
6
6𝑥𝑥 + 3(2) = 12
6𝑥𝑥 + 6 = 12
6𝑥𝑥 = 6
=6
𝑥𝑥 = 1
6
−6 − 6
Therefore, our answer is (1,2) which happens to be the point of intersection.
East Campus, CB 117
361-698-1579
Math Learning Center
West Campus, HS1 203
361-698-1860
Method 4: TI-83 / TI- 84 Matrix Solution
GRAPHING CALCULATOR
GUASS-JORDAN METHOD (reduced-row echelon form)
1. Write the augmented matrix
2. 2nd x-1 for MATRIX* > > EDIT, 1: [A]
3. Input the number of rows and then the number of
columns, input the values from the augmented matrix
4. 2nd MODE to QUIT
5. MATRIX > MATH, Choose: B: rref(
6. MATRIX, NAMES, Choose: 1: [A]
Example:
Solve the system.
x + y – 4z = 10
2x – 3y + z = 7
3x – y – z = 12
The augmented
matrix is:
1 1 −4 10
�2 −3 1
7�
3 −1 −1 12
Define the size of the
matrix. Rows by columns.
*Press 2nd x-1 for
MATRIX and scroll
over to EDIT
*Hit Enter
Now the screen looks
like this. We still need
to input the name of
the matrix.
Go to MATRIX and
scroll over to MATH
To leave this screen press
2nd MODE which is QUIT
Go to MATRIX
and select 1: [A]
Scroll down to
option B: rref(
Now hit ENTER
This last matrix corresponds to the system:
x=3
y = -1
z = -2
TADA!
* On the TI-83 the MATRX button is
used rather than 2nd x-1 for MATRIX
Answer: {(3, -1, -2)}
Turn decimals into fractions by going to MATH, then select
option 1: > Frac, and then ENTER.