Solving Systems of Linear and Quadratic Equations
Solving Systems of Linear and Quadratic Equations
2 8
10
x y
x y
Systems of Linear Equations
We had 3 methods to solve them.
Method 1 - Graphing
Solve for y.
2 8y x
10y x (6, – 4)
Systems of Linear Equations
Method 2 - Substitution
2 8
10
x y
x y
2 8y x
10x y 10x
2 8x
2 8 10x x 3 18x
6x 6
2 8y x 2( ) 8y
12 8y 4y
(6, – 4)
Systems of Linear Equations
Method 3 - Elimination
2 8
10
x y
x y
2 8
10
x y
x y
3 18x
6x 6
10x y ( ) 10y
4y 4y
(6, – 4)
Try the following.
1 2,
2 3
(12,1)1.
2.
3.
4.
5.
6.(12, – 4)
Many solutions
Same Line!
1 1,
6 8
NO solution
Parallel Lines!
Solve the System AlgebraicallyUse Substitution
1
12
xy
xy
1 xy( ) 1x
Answer: (0,1) (1,2)
2 1x
02 xx
1 0x x
0 1 0x or x
0 1x or x
0x
20 1
1
y
y
(0, 1)
1x
21 1
2
y
y
(1, 2)
Solve the System AlgebraicallyUse Substitution
2 2y x 2( ) 2 2x x 22 x
Answer: (2,2)
2 2 2y x x
2x
(2, 2)
22
222
xy
xxy
20 4 4x x
0 2 2x x
0 2x 2x
2 2 2
4 2
2
y
y
y
Solve the System AlgebraicallyUse Substitution
6y x
22 6 26x x
2 2 12 36 26x x x
6
2622
yx
yx
22 12 10 0x x 2 6 5 0x x
5 1 0x x
5 1x or x
5x
5 6
1
1
y
y
y
1x
1 6
5
5
y
y
y
(5, –1)
(1, – 5)
Answer: (5, –1) (1, – 5)
Solve the System AlgebraicallyUse Substitution
3
4y x
2
2 325
4x x
2 2925
16x x
2 216 9 400x x 225 400x
2 16x
4x
4x
4 3 4
4 12
3
y
y
y
4x
4 3 4
4 12
3
y
y
y
(4, 3)
(– 4, – 3)
Answer: (4, 3) (–4, – 3)
xy
yx
34
2522
2 2
2 2
3 4 11
13
x y
x y
Solve the System AlgebraicallyUse ????
Now let’s look at the Graphs of these Systems!
Classify each equation
as linear/quadratic.
What does the graph
of each look like?
Line
Parabola
Linear
Quadratic
What is the solution
to the system?Point of Intersection
(-2, 0)Point of Intersection
(1, -3 )
1
12
xy
xy
Answer:
(0,1) (1,2)
22
222
xy
xxy
Answer:
(2,2)
Now use your calculator
to check it graphically.
We have solved the following algebraically
Equations must be solved for y!!
6
2622
yx
yx
2 226y x
2 2 26x y
226y x
6
and
y x
The first equation can be entered as two separate entries:
or a "list" notation may be used:
Zoom 6: ZStandardZoom 5: ZSquare
Answer: (5, –1) (1, – 5)