Solve for “y” for each: 1.) 2x + y = 9 2.) -x = y + 3 3.) 2y – 4 = 3x 4.) 2 = y – ⅓x 5.) 7x – y = -8 6.) 3y – 1 = -3x -2x -2x - 3 - 3 y = -2x + 9 y = -x - 3 +4 +4 +⅓x +⅓x 2y = 3x + 4 y = ⅓x + 2 -7x -7x +1 +1 -y = -7x - 8 y = 7x + 8 3y = -3x + 1

Again, solve for “y” 1.) 2x + 2y = 10 2.) 6x – 3y = -3 3.) 2y – 7 = -x 4.) 15x + 5y = 20 - 2x - 2x - 6x - 6x -3y = -6x - 3 2y = -2x + 10 Divide by 2 Divide by -3 -3 -3 -3 y = -x + 5 y = 2x + 1 + 7 + 7 - 15x - 15x 2y = -1x + 7 5y = -15x + 20 Divide by 2 Divide by 5 y = -½x + 3.5 y = -3x + 4

3.3 Graphing Linear Equations

Linear Equation An equation for which the graph is a STRAIGHT line NO FRACTIONS and NO LEADING NEGATIVES!!!

Linear Equation: an equation whose graph forms a line. In linear equations, all variables are taken to the first power. is linear. Why? is not.

What is a Linear Equation? y x Linear A linear equation is an equation whose graph is a LINE. Not Linear

Solution INFINITE AMOUNT OF SOLUTIONS (x,y) Any ordered pair of numbers that makes a linear equation true. FOR: y = x - 9 (9,0) IS ONE SOLUTION (x,y) Can you think of another solution? Is (11,2)?? INFINITE AMOUNT OF SOLUTIONS Yes Is (-5,-14)?? Yes

Determine if the ordered pair given is a solution of x + 2y = 5 1.) (1,2) 2.) (7,-3) (1) + 2(2) = 5?? (7) + 2(-3) = 5?? 5 = 5 1 = 5 NO YES

Steps to GRAPHING 1.) SOLVE FOR “y” 2.) MAKE A T-CHART 3.) GRAPH EACH ORDERED PAIR 4.) CONNECT THE DOTS TO DRAW THE LINE

Linear Equation Example: y = x + 3

Find solutions using table Graphing Find solutions using table y = x + 3 x y -2 -1 1 2 3 x y -2 1 -1 2 3 4 5 6 Create these ordered pairs (-2,1) and (-1,2) and (0,3) and (1,4) and (2,5) and (3,6)

Graphing Graph the points from the table (-2,1) (-1,2) (0,3) (1,4) (2,5)

Graphing Draw a line to connect them So by looking at this line we can assume that (5,8) is a solution? What other points can we conclude is a solution??? y = x +3

Then creating a table of values. Suppose you were asked to graph the solution set of the linear equation, x + y = 3 -x -x y = -x + 3 FIRST SOLVE FOR Y!! Then creating a table of values.

Use the x-values -2, -1, 0, 1, and 2 -2 -1 0 1 2 5 4 3 2 1

With your table completed for three sets of values, you are ready to graph the line. y = -x + 3 -2 -1 0 1 2 5 4 3 2 1

Graphing a Linear Equation y x How do we graph linear equations? Let’s try this one: y = 3x – 2 Make a Table of values x y –2 –1 1 2 –8 y = 3(–2) – 2 = –8 –5 y = 3(–1) – 2 = –5 –2 y = 3(0) – 2 = –2 1 y = 3(1) – 2 = 1 4 y = 3(2) – 2 = 4

Graphing a Linear Equation y x How about another one! Let’s try x – 2y = 6. First Step: Solve for y x – 2y = 6 –2y = -x + 6 y = ½x - 3 Second Step: Make a t-chart x y –4 –2 2 4 -5 ½(-4) – 3 = -5 -4 ½(-2) – 3 = -4 -3 ½(-0) – 3 = -3 -2 -1

Create a table of values for each equation, and graph it on grid paper: 1. x y -2 3 -1 4 5 1 6 2 7 8 2. x y -2 11 -1 9 7 1 5 2 3 y = x + 5 y =-2 x + 7 3. x y -2 -8 -1 -6 -4 1 2 3 x y -4 -1 -2 1 2 4 3 6 4.

Make a table for each function using x-valuse -2, -1, 0, 1, 2, 3: Try These Make a table for each function using x-valuse -2, -1, 0, 1, 2, 3: y = 2x + 3 2) y = -x - 4 x y -2 -1 1 3 5 2 7 9 x y -2 -1 -3 -4 1 -5 2 -6 3 -7

Find some solutions of this linear equations 1.) -2x + y = -3 First Solve for y y = 2x - 3 x y -1 -5 -3 1 2 3 Use the x-values of -1, 0, 1, 2 and 3

Make a t-chart for this function: 1.) -4x + y = -1

Graph the function y = 3x -2 First make a t-chart

Graph the function: 4y – 2x = 8 First solve for “y” 4y – 2x = 8 +2x +2x 4y = 2x + 8 4 4 y = ½x + 2 x y -4 -2 1 2 3 4 6 5 Then use which x-values???

Graph the function: 3x – 2y = 6 First solve for “y” 3x – 2y = 6 -3x -3x -2y = -3x + 6 -2 -2 -2 y = 3/2 x - 3 x y -4 -9 -2 -6 -3 2 4 3 6 Then use which x-values???

Graph the equation y = 2x + 6

Graph the equation: -6x + 2y = 4 y = 3x + 2 Plot (-2, -4), (0, 2) and (1, 5) Then draw the line. Make sure your line covers the graph and has arrows on both ends. Be sure to use a ruler.

Which ordered pair does NOT satisfy the equation 2x + y = 7? Which ordered is a solution of the equation 5x – 3y = 17? a.) (4,1) b.) (1,4) c.) (5,1) d.) (1,5) a.) (1,5) b.) (3,2) c.) (0,7) d.) (4,-1)