6.1 Solving Linear Systems by Graphing

Standard: SWBAT solve a system of two linear equations in two variables and are able to interpret the answer graphically.

http://apod.nasa.gov/apod/ap081212.html

Lick Observatory Moonrise

Mini Quiz 48 Is the ordered pair a solution to the equation 2x – 3y = 5 1.

(1, 0) 2.

3.

(4, 1) Graph the line 2x – y + 3 = 0

Overview  Solving Systems of Linear Equations (two equations) by Graphing  Understanding No Solution  Understanding Infinite Solution

Graphing Lines Quick Review What are the different ways to graph a line?

 T-chart  Slope-Intercept Form:

y

= m

x

+ b  x- and y-intercepts: Using Standard Form: A

x

+ B

y

= C Graph the line

x

– 2

y

– 4 = 0

Graphing Systems of Linear Equations Step:

1.

Graph each line

plane on the same coordinate

2.

3.

Label the point

(ordered pair) where the lines cross (intersect)

Check

your answer (YES, you need to check both equations!)

Graphing Lines Solve the Systems by Graphing 1.

y

 3

x

 1

y

  2

x

 4 Check: (1, 2)

y

= 3

x

– 1 2 = 3( 1 ) – 1 2 = 3 – 1 2 = 2 

y

= -2

x

+ 4 2 = -2( 1 ) + 4 2 = -2 + 4 2 = 2  Where do the lines cross?

(1, 2)

Graphing Lines Solve the Systems by Graphing 2.

y

 4

x

 7

y

  3

x

Where do the lines cross?

(1, -3) Check: (1, -3)

y

= 4

x

– 7 -3 = 4( 1 ) – 7 -3 = 4 – 7 -3 = -3 

y

= -3

x

-3 = -3( 1 ) -3 = -3 

Graphing Lines Solve the Systems by Graphing 3.

y

 2

x

 7

y



x

 6 Check: (-1, 5)

y

= 2x + 7 5 = 2( -1 ) + 7 5 = -2 + 7 5 = 5 

y

= x + 6 5 = ( -1 ) + 6 5 = 5  Where do the lines cross?

(-1, 5)

Special Cases – No Solution and Infinitely Many Solutions No Solutions 4.

y

 2

x

 6 4

x

 2

y

 8 Same slope

y

 2

x

 6

y

 2

x

 4 6.

x

Infinitely Many Solutions 

y

 4

y

 

x

 4 2

x

 2

y

 8 Same slope

y

 

x

 4 Different y-intercept 5.

x

2

x

  2

y

4

y

 Solve for 

y

10 10

y y

    1 2 1 2

x



x

 5 5 2 7.

Same y-intercept 3

x

 5

y

 0

y

 3

x

5 3

y



x

 3 5

y

5

x

and look at the equations, what do you notice about the equations?

Graphing Lines Solve the Systems by Graphing 8.

y

 

x

 2 3

x

 3

y

 12

x

-intercept (

x

, 0) 3

x

= 12

x

= 4 (4, 0)

y

-intercept (0,

y

) 3

y

= 12

y

= 4 (0, 4) Where do the lines cross?

No Solution

Graphing Lines Solve the Systems by Graphing 9.

y

  3

x

 3 3

x



y

  3 Where do the lines cross?

x

-intercept (

x

, 0) Infinitely Many Solutions 3

x

= -3

x

= -1

y

-intercept (0,

y

)

y

= -3 (0, -3)

Graphing Lines Solve the Systems by Graphing 10. 2

x



y

 0

y

  2

x

 4 Slope-intercept form 2

x

+

y =

0

y

= -2

x

Where do the lines cross?

No Solution

Application 11. Suppose you have $20 in your bank account. You start saving $5 each week. Your friend has $15 in her account and is saving $10 each week. When will you and your friend have the same amount of money in your account?

y = 5x + 20 y = 10x + 5 After 3 weeks!

35 30 25 20 15 10 5 1 2 3 4 5 6 Number of Weeks

Wrap Up  Solving Linear System by Graphing  Graph each line  Find where they cross  Check solution HW: P. 279 #13-23 odd, P. 281 #43-51 odd

DLUQ: When graphing a linear system, where do you find the solution?