solving systems of equations graphically equations on the coordinate plane we moved on to solving...

Unit 5: Graphs of Systems and Inequalities

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Solving Systems of Equations Graphically

A system of equations is a collection of two or more equations with a same set of unknowns. In

solving a system of equations, we try to find values for each of the unknowns that will satisfy every

equation in the system. When solving a system containing two linear equations there will be one

ordered pair (x,y) that will work in both equations.

To solve such a system graphically, we will graph both lines on the same set of axis and look for the

point of intersection. The point of intersection will be the one ordered pair that works in both

equations. We must then CHECK the solution by substituting the x and y coordinates in BOTH

ORIGINAL EQUATIONS.

1) Solve the following system graphically:

y = 2x – 5

y = - ⅓x + 2


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Solution (3,1)


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Solve each of the systems of equations graphically:

2) y + 1 = -3(x – 1) 7x + 7y = 42


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3) y – 9 = ¾ (x – 12) 6x + 12y = -60


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4) 12x – 8y = 48

y – 4 = -2(x – 2)


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5) y + 5 = 2(x + 4) y – 10 = - ½ (x + 4)


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Solve each system graphically and check:

6) y = -4x -5 10) y-2= (3/5)(x-10)

y = 2x -7 y+11 =2(x+7)

7) 6x + 3y =21 11) 6x + 9y = 45

12x + 16y = -48 9x +15y = 75

8) 12x – 6y = -6 12) x = 5

16x -8y = 40 y-12 = -3(x+2)

9) y= -4

x = 7


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1) 36x – 12 y = 24 2) y – 8 = ½ (x – 2)

y - 4 = ⅓(x + 6) 24x + 24y = 24

3) y = -5 4) 27x + 18y = 72

y + 7 = -2(x – 2) y + 5 = -3(x – 2)

5) 15x + 30y = -120 6) 16x – 32y = -128

y - 1 = (3/2)(x + 2) 18x + 24y = -24


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Graphing Inequalities

When we solved and graphed inequalities with only one variable (ex: x > 3), we moved on

to compound inequalities (AND/OR). We would graph both inequalities on the same number line

and decide what to keep based on whether it was an AND or an OR problem. When we graphed

linear equations on the coordinate plane we moved on to solving systems of equations graphically.

When we graph inequalities in two variables on the coordinate plane, we do not graph compound

inequalities. We move on to solving systems of inequalities. It takes a little from both inequalities

with one variable and solving systems graphically.

Graph the Inequality:

y > ¼ x + 3

Step 1: Graph the line.

y > ¼ x + 3

m = ¼ = ▲y = up 1

▲x right 4

y-int= (0,3)

Step 2: Test a point one up from the from

the y-int and one down from the y-int):

(0, 2) (0, 4)

2 > ¼ (0)+3 4 > ¼ (0) + 3

2 > 3 4 > 3

FALSE TRUE

Step 3: Shade towards the “true” point (0,4)

When you “test”, you must do it in the original equation!


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1) 6x - 9y > 36

2) y - 3 > -2(x + 1)


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3) 12x + 9y < 27

4) y + 4 > -3(x - 3)


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5) y > 4

6) x < -6


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Graphing Systems of Inequalities

Solve the system of inequalities graphically: y > ¼ x + 3

y < 3x – 5

Step 1: Graph the 1st inequality (graph and test a point one up from the y-int and one down from the

y-int.):

y > ¼ x + 3

m = ¼ =

y-int= (0,3)

(0, 2) TEST (0, 4)

2 > ¼ (0)+3 -3 > ¼ (0) + 3

2 > 3 4 > 3

FALSE TRUE

Step 2: Graph the 2nd

inequality (graph the line and test a point one up from the y-int and one down

from the y-int.):

y < 3x – 5

m = 3/1 =

y-int.= (0,-5)

(0-6) TEST (0,-4)

-6 < 3(0) - 5 -4 < 3(0) - 5

-6 < -5 -4 < -5

TRUE FALSE


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Step 3: Label the area where the shading intersects with an “S”

2) y - 3 < - ⅓(x – 6)

12x – 6y > -12


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3) x > 4

y < -5

4) 24x + 6x > -6

y > 2


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5) y – 6 < ⅔(x - 9)

x < -3

solving systems of equations graphically equations on the coordinate plane we moved on to solving...

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