f(x)= -1/2x2 + 18
f(x)= 2x2 - 2
Solving
Systems of Equations
by Fen Xu and Timothy Lou Ly
The Concept
Graphing a Linear Equation
(12, 20)
(10, 16)
(8, 12)
y = 1x
X-Values Y-Values
-24
-20
-16
-12
-8
-24
-20
-16
-12
-8
y = 2x - 4
X-Values Y-Values
(-8, -8)
(-16, -16)
(-20, -20)
8
9
10
11
12
12
14
16
18
20
The Concept
Dependent System
Inconsistent System
Two overlapping lines with
the same slope and points
Two lines with the same slopes
that never intersect or share points.
The Concept
Three Ways to Solve
- Graphing
- Addition
- Substitution
Addition
Step 1: Make equations into simplest form of
Ax + By = C
C.
_
_
6(
x - 2y = 5
x-1 y+2
+
=4
2
3
)
Multiply by LCD to get rid of fractions
= 3x - 3 + 2y + 4 = 24
Combine like terms by adding -3
and +4 together
= 3x + 2y - 1 = 24
Add +1 to both sides to cancel -1
and isolate variables
= 3x + 2y = 23
Step 2: Choose to solve for “x” or “y.” For the example,
we’ll solve for “x.”
Addition
Step 3: Add the two equations so that the y-values
cancel.
Because
2y
is
being
subtracted
from
2y,
they
3x + 2y = 23
/
+ ( x - 2y
=
5)
/
4x = 28
cancel
If you wanted to solve for “y” and cancel
“x,” you would need to multiply the 2nd
equation by -3
Step 4: Continue to solve for “x.”
4
4x = 28
x=7
Divide both sides by 4
Addition
Step 5: We can now put 7 in for “x” in any equation
and find the value of “y.”
x - 2y = 5
CHECK: x- 2y = 5
3x + 2y = 23
= 7 - 2y = 5
(7) - 2(1) = 5
3(7) + 2(1) = 23
= -2
-2y = -2
(7) - 2 = 5
21 + 2 = 23
=y=1
5=5
√
23 = 23
√
That’s it! (7, 1) is your solution/intersection!
Substitution
Step 1: Solve for one of the variables from one equation
equation.
x - 2y = 5
_
_
6(
x-1 y+2
+
=4
2
3
)
Multiply by LCD to get rid of fractions
= 3x - 3 + 2y + 4 = 24
Combine like terms by adding -3
and +4 together
= 3x + 2y - 1 = 24
Add +1 to both sides to cancel -1
and isolate variables
= 3x + 2y = 23
Subtract 3x from both sides
=2
2y = -3x + 23
Divide the equation by 2
_
-3
x + 11.5
=y=
2
Substitution
-_
3
x + 11.5 ” for “y” in the other
Step 2: Substitute “ y =
2
equation: x - 2y = 5 and solve.
_
-3
x - 2( x + 11.5) = 5 Multiply out -2
2
Add 23 to both sides
= x + 3x - 23 = 5
=4
4x = 28
Divide the equation by 4
=x=7
Step 3: Putting in 7 for “x,” we know that y = 1
1.
x - 2y = 5
= 7 - 2y = 5 = -2
-2y = -2 = y = 1
_
-3
y = x + 11.5
2
(0, 11.5)
(4, 5.5)
X-Values
Y-Values
0
2
4
6
7
11.5
8.5
5.5
2.5
1
_
1 x - 2.5
y=
2
(4, -0.5)
(0, -2.5)
X-Values
Y-Values
0
2
4
6
7
-2.5
-1.5
-0.5
0.5
1
Review
Try to solve this with the method of your choice:
7x-6y=-6
-7x+6y=-4
We’ll check, and if you get it right, you get some candy!
(You’ll all get some anyways)