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)