Solving Systems of Equations
3 Approaches
Ms. NongAdapted from Mrs. N. Newman’s PPT
Click here to begin
Method #1
Graphically
Method #2
Algebraically Using Addition and/or Subtraction
Method #3
Algebraically Using Substitution
POSSIBLE ANSWER:
Answer: (x, y) or (x, y, z)
Answer: No Solution Answer: Identity
In order to solve a system of equations graphically you
typically begin by making sure both equations are in Slope-
Intercept form.
Where m is the slope and b is the y-intercept.
Examples:
y = 3x- 4
y = -2x +6
Slope is 3 and y-intercept is - 4.
Slope is -2 and y-intercept is 6.
bmxy
Looking at the System Graphs:
•If the lines cross once, there will be one solution.
•If the lines are parallel, there will be no solutions.
•If the lines are the same, there will be an infinite number of
solutions.
In order to solve a system of equations algebraically using addition first you must be sure that both equation are in the same chronological order.
Example: 2
4
yx
xy
2
4
xy
xyCould be
Now select which of the two variables you want to eliminate.
For the example below I decided to remove x.
2
4
xy
xy
The reason I chose to eliminate x is because they are the additive inverse of each other.
That means they will cancel when added together.
Now substitute the known value into either one of the original equations.I decided to substitute 3 in for y in the second equation.
1
23
x
x
Now state your solution set always remembering to do so in alphabetical order.
[-1,3]
Lets suppose for a moment that the equations are in the same sequential order. However, you notice that neither coefficients are additive
inverses of the other.
1273
332
yx
yx
Identify the least common multiple of the coefficient you chose to
eliminate. So, the LCM of 2 and 3 in this example would be 6.
Multiply one or both equations by their
respective multiples. Be sure to choose numbers that
will result in additive inverses.
)1273(2
)332(3
yx
yx
24146
996
yx
yxbecomes
Now substitute the known value into either one of the original equations.
3
62
392
3)3(32
3
x
x
x
x
y
In order to solve a system equations algebraically using substitution you must have
one variable isolated in one of the equations. In other words you will need to
solve for y in terms of x or solve for x in terms of y.
In this example it has been done for you in the first
equation.
2
4
yx
xy
Now lets suppose for a moment that you are given a set of equations like this..
1273
332
yx
yx
Choosing to isolate y in the first equation the result is :
13
2 xy
Now substitute what y equals into the second equation.
2
4
yx
xy
becomes24 xx
Better know as
Therefore 1
22
242
x
x
x
y = 4x3x + y = -
21Step 5: Check the solution in both equations.
y = 4x
-12 = 4(-3)
-12 = -12
3x + y = -21
3(-3) + (-12) = -21
-9 + (-12) = -21
-21= -21
Top Related