1. Solving systems of equations by graphing
Lesson 55
Solving systems of equations by graphing

2. System of linear equations
A system of linear equations consists of 2 or more linear equations containing 2 or more variables.
Example: 3x+ y = 9
x + 2y = 8

3. Solution of a system of linear equations
The solution to a system of linear equations is any ordered pair that makes all equations true

4. Identifying solutions
Tell whether the ordered pair is a solution to the system of equations.
( 1,2) for 3x + 2y = 7
x = 7 -3y
Replace x and y in both equations to see if the equation is true.
3(1) + 2(2) = 7
= 7 true
1 = 7 - 3(2)
1 = true
(1,2) is a solution because it makes both equations true.

5. practice
Tell whether the ordered pair is a solution to the given systems.
(2,3) x = 4y - 10
3x + 7y = 27
(1,5) y = 4x + 1
y-3x + 4
(2,3) x = 7 - 3y
3x + 2y = 12

6. Solutions to systems of linear equations
If a system of linear equations has 1 solution , it is the point at which the 2 lines intersect on the coordinate plane.

7. Solving by graphing
All solutions of a linear equation can be found on its graph.
Graphing each equation can solve a system of linear equations.
If the system has 1 solution, the solution is the common point or the point of intersection.

8. Solving by graphing Solve by graphing and check your solution
y = x + 3
y = 2x + 1
y = x + 5
y = 3x -1
y = 2x - 5
y = x - 3

9. Writing equations in slope-intercept form
Solve by graphing and check your solution. First, write each equation in slope-intercept form.
3x + y = y = -3x + 9
x + 2y = 8 2y = -x y = -1/2 x + 4
4x + 3y = 1
7x + y = 6
y = 2x -5
3x + y = 15

10. Solving using a graphing calculator- LAB 5
Enter the equations into the Y= Editor
Graph the equations in the standard window ( ZOOM:Standard)
To find the approximate point of intersection, TRACE-use the arrow keys to get the cursor as close as possible to the point of intersection.
To calculate the EXACT intersection, press 2nd CALC and select 5:Intersection
At the prompt "First Curve?", press ENTER
At the next prompt "second curve?" press ENTER
Use the arrow keys to move the cursor close to the point of intersection.
At the prompt "guess?" , press ENTER
The solution will be at the bottom of the screen

11. Changing decimal coordinates to fraction coordinates
Press x and ENTER, and then press MATH and select 1:>FRAC.
Press ENTER.
Press ALPHA [Y] and ENTER, and then press MATH and select 1: >Frac
Press Enter
This will change your decimal answers to fractions.

12. practice solve by using the TRACE function. y = -2x + 3 y = .5x + 1
Use the intersection feature to find the exact intersection
Use intersection feature to find the exact intersection of y = -3/2 x -5
y = 1/3 x + 5
Answers should be in fraction form

13. Transforming linear functions
Investigation 6
Transforming linear functions

14. Family of functions A family of functions share common characteristics
A parent function is the simplest function in a family of functions.
The parent function for a linear equation is
f(x) = x
All other linear equations are transformations of the parent function

15. translation
A translation shifts every point in the figure the same distance in the same direction. A translation can be thought of as a slide. It is also referred to as a vertical change
Graph y = x
y = x + 3
y = x - 2
Compare the graphs

16. Vertical stretch or compression
A vertical stretch or compression changes the rate of change ( slope) of a linear equation.
Graph y= x
y = 3x
y = 5x
y = 1/3 x
What happens to the line?
How is the slope related to the steepness of the line?

17. reflection
A reflection produces a mirror image across a line. It can be thought of as a flip.
Graph y = 2x
y = -2x
y = 1/2 x
y = -1/2 x
How is the slope related to the reflection of a line?

18. More than one transformation
Graph f(x) = x
Graph the following and describe the transformation
f(x) = x + 4
f(x) = -x
f(x) = 1/2 x
f(x) = 4x -2