Graphing Linear Equations Objectives: …to check and find possible solutions to equations with two variables ...to graph solutions to linear equations on the coordinate plane

Assessment Anchor:

8.C.3.1 – Plot and/or identify ordered pairs on a coordinate plane. 8.D.4.1 – Represent relationships with tables or graphs on the coordinate plane.

NOTES (checking) To check if an ordered pair is a solution to an equation containing two variables (x and y): 1. Substitute the “x” and “y” values into the equation for “x” and “y” respectively 2. Simplify each side of the equation as far as possible 3. Determine if the numerical statement is TRUE or FALSE. If it is TRUE, then the ordered pair is a solution.

EXAMPLES 1a) Is the ordered pair (2, 1) a solution to: Substitute (2, 1) IS a solution!

1b) Is the ordered pair (-1, 3) a solution to: Substitute (-1, 3) is NOT a solution!

4x + 2y = 10 4(2) + 2(1) = 10 8 + 2 = 10 10 = 10 TRUE!

4x + 2y = 10 4(-1) + 2(3) = 10 -4 + 6 = 10 2 = 10 FALSE!

Graphing Linear Equations 1c) Is the ordered pair (5, -5) a solution to:

4x + 2y = 10

Substitute

2a) Is the ordered pair (-4, -1) a solution to: Substitute (-4, -1) is NOT a solution!

2b) Is the ordered pair (1, -3) a solution to:

-2y = 5x – 11 -2(-1) = 5(-4) – 11 2 = -20 – 11 2 = -31 FALSE

-2y = 5x – 11

Substitute

“Remember, there are an infinite number of ordered pairs that will be solutions to an equation with two variables.”

MORE NOTES (finding) To find a solution to an equation containing two variables (x and y): 1. Choose a value for “x”. You may choose any number you wish. 2. Substitute that value into the equation for “x”, and solve for “y” 3. Write your solution as an ordered pair (x,y)

Graphing Linear Equations EXAMPLES 3)

4)

Find 3 solutions to:

4x + 2y = 10

1st solution

2nd solution

3rd solution

Choose… x = 3

Choose… x = -5

Choose… x = 6

4x + 2y = 10 4(3) + 2y = 10 12 + 2y = 10 – 12 – 12 2y = -2 2 2 y = -1

4x + 2y = 10 4(-5) + 2y = 10 -20 + 2y = 10 + 20 + 20 2y = 30 2 2 y = 15

4x + 2y = 10 4(6) + 2y = 10 24 + 2y = 10 – 24 – 24 2y = -14 2 2 y = -7

1st solution: (3, -1)

2nd solution: (-5, 15)

3rd solution: (6, -7)

Find 3 solutions to:

3y = 8x + 1

1st solution

2nd solution

3rd solution

Choose…

Choose…

Choose…

Graphing Linear Equations Vocabulary alert!! LINEAR EQUATION – an equation whose graph is a straight line

EVEN MORE NOTES (graphing) To graph linear equations (using T-tables): 1. Make a T-table that contains at least 3 ordered pairs a. Choose whatever numbers you want for “x”, but keep it simple b. Substitute a value for “x”, solve for “y” and fill in the table 2. Make the graph a. The straight line shows all possible solutions to the equation b. If points don’t make straight line, double check the calculations in step #1 and the plotting of ordered pairs c. Don’t forget to include arrows on your line!

COORDINATE PLANE

y-axis

Quadrant II

Quadrant I

Quadrant III

Quadrant IV

x-axis

origin

Graphing Linear Equations EXAMPLES 5)

Graph this equation:

y = 3x – 2

T-Table Use x = 1 x 0 1 2

y -2 1 4

y = 3(1) – 2 y=3–2 y=1

Use x = 0

Use x = 2

y = 3(0) – 2 y=0–2 y = -2

y = 3(2) – 2 y=6–2 y=4

6)

Graph this equation:

Solve for “y” first:

4y = -8x + 20 4 4 y = -2x + 5

T-Table Use x = 1 x 0 1 2

y 5 3 1

y = -2(1) + 5 y = -2 + 5 y=3

Use x = 0

Use x = 2

y = -2(0) + 5 y=0+5 y=5

y = -2(2) + 5 y = -4 + 5 y=1

4y = -8x + 20 Now graph: y = -2x + 5

Graphing Linear Equations 7)

Graph this equation:

y= x+4

Graph this equation:

2y – 8 = -6x

T-Table x y

8)

Solve for “y” first:

T-Table x y

Assessment Anchor:

8.C.3.1 – Plot and/or identify ordered pairs on a coordinate plane. 8.D.4.1 – Represent relationships with tables or graphs on the coordinate plane.

NOTES (checking) To check if an ordered pair is a solution to an equation containing two variables (x and y): 1. Substitute the “x” and “y” values into the equation for “x” and “y” respectively 2. Simplify each side of the equation as far as possible 3. Determine if the numerical statement is TRUE or FALSE. If it is TRUE, then the ordered pair is a solution.

EXAMPLES 1a) Is the ordered pair (2, 1) a solution to: Substitute (2, 1) IS a solution!

1b) Is the ordered pair (-1, 3) a solution to: Substitute (-1, 3) is NOT a solution!

4x + 2y = 10 4(2) + 2(1) = 10 8 + 2 = 10 10 = 10 TRUE!

4x + 2y = 10 4(-1) + 2(3) = 10 -4 + 6 = 10 2 = 10 FALSE!

Graphing Linear Equations 1c) Is the ordered pair (5, -5) a solution to:

4x + 2y = 10

Substitute

2a) Is the ordered pair (-4, -1) a solution to: Substitute (-4, -1) is NOT a solution!

2b) Is the ordered pair (1, -3) a solution to:

-2y = 5x – 11 -2(-1) = 5(-4) – 11 2 = -20 – 11 2 = -31 FALSE

-2y = 5x – 11

Substitute

“Remember, there are an infinite number of ordered pairs that will be solutions to an equation with two variables.”

MORE NOTES (finding) To find a solution to an equation containing two variables (x and y): 1. Choose a value for “x”. You may choose any number you wish. 2. Substitute that value into the equation for “x”, and solve for “y” 3. Write your solution as an ordered pair (x,y)

Graphing Linear Equations EXAMPLES 3)

4)

Find 3 solutions to:

4x + 2y = 10

1st solution

2nd solution

3rd solution

Choose… x = 3

Choose… x = -5

Choose… x = 6

4x + 2y = 10 4(3) + 2y = 10 12 + 2y = 10 – 12 – 12 2y = -2 2 2 y = -1

4x + 2y = 10 4(-5) + 2y = 10 -20 + 2y = 10 + 20 + 20 2y = 30 2 2 y = 15

4x + 2y = 10 4(6) + 2y = 10 24 + 2y = 10 – 24 – 24 2y = -14 2 2 y = -7

1st solution: (3, -1)

2nd solution: (-5, 15)

3rd solution: (6, -7)

Find 3 solutions to:

3y = 8x + 1

1st solution

2nd solution

3rd solution

Choose…

Choose…

Choose…

Graphing Linear Equations Vocabulary alert!! LINEAR EQUATION – an equation whose graph is a straight line

EVEN MORE NOTES (graphing) To graph linear equations (using T-tables): 1. Make a T-table that contains at least 3 ordered pairs a. Choose whatever numbers you want for “x”, but keep it simple b. Substitute a value for “x”, solve for “y” and fill in the table 2. Make the graph a. The straight line shows all possible solutions to the equation b. If points don’t make straight line, double check the calculations in step #1 and the plotting of ordered pairs c. Don’t forget to include arrows on your line!

COORDINATE PLANE

y-axis

Quadrant II

Quadrant I

Quadrant III

Quadrant IV

x-axis

origin

Graphing Linear Equations EXAMPLES 5)

Graph this equation:

y = 3x – 2

T-Table Use x = 1 x 0 1 2

y -2 1 4

y = 3(1) – 2 y=3–2 y=1

Use x = 0

Use x = 2

y = 3(0) – 2 y=0–2 y = -2

y = 3(2) – 2 y=6–2 y=4

6)

Graph this equation:

Solve for “y” first:

4y = -8x + 20 4 4 y = -2x + 5

T-Table Use x = 1 x 0 1 2

y 5 3 1

y = -2(1) + 5 y = -2 + 5 y=3

Use x = 0

Use x = 2

y = -2(0) + 5 y=0+5 y=5

y = -2(2) + 5 y = -4 + 5 y=1

4y = -8x + 20 Now graph: y = -2x + 5

Graphing Linear Equations 7)

Graph this equation:

y= x+4

Graph this equation:

2y – 8 = -6x

T-Table x y

8)

Solve for “y” first:

T-Table x y