Chapter 3

3.1: Solving Linear Systems by GraphingGroup 4

Goal

Get two variables, (x,y), to correctly come out of two equations

ax+by=c dx+ey=f

Check whether the ordered pair is a solution of the system.

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Checking Solutions of Linear Equations

Example 1: Is (2,2) and (0,1) solutions of:

3x-2y=2X+2y=6

Put variables (2,2) into equations:3 (2)-2 (2)=2*2 + 2 (2)=6*

Because both variables solve both equationsIt is a Solution of the System

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Checking Solutions of Linear System

Example 2:Put variables (0,1) into equations: 3(0)-2(1)=2 *0+2(1)=-2≠6

Because Equation 2 does not check It is not a Solution of the System

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Solving A System Graphically

Goal: To solve a system of equations graphically, graph both equations and see where they intersect.  The intersection point

is the solutionExample 1: 4x-6y=12 2x+2y=6Solve for y= mx+b to graph 4x-6y=12 2x+2y=6 4x=6y+12 2y=-

2x+6 4x-12=6y y= x+ 6y=4x-12 y=-x+3 y= x- y= x-2

Graph y= x-2 and y=-x+3

6

4

6

12

3

2

3

2

2

22

6

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Solution

•The point of intersection of the two lines (3,0) is the solution to the system of equations.•This means that (3,0), when substituted into either equation, will make them both true.•Use the check method to check your answer.

4x-6y=12 4(3)-6(0)=12 2x+2y=6 2(3)+2(0)=6

Solution of Systems

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Systems with Many or No Solutions

Systems with Many SolutionsExample 1:

3x-2y=6

6x-4y=12 Graph the equation (Solving each

equation for y).

3x-2y=6 y= x-3

6x-4y=12 y= x-3 The graph of the equation is the

Same line.

So Infinitely there are many solutions.

2

3

2

3

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Examples

Systems with No SolutionsExample 2:

-x+5y=8 2x-10y=7

Graph the equations solving for y.

The lines are parallel and do not intersect

So , There is no solution.

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Review Examples

Checking Solutions of Linear Equations

Is the ordered pair (1, 3) a solution of the given system of equations?

3x + 5y = 18 x – 3y = –8

3x+5y=18X-3y=-8

3(1) + 5(3) = 183 + 15 = 1818 = 18

1 – 3(3) = –81 – 9 = –8–8 = –8Solution of Equations

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Review

Solving a System of Equations Graphically

x – y = 43x + y = 0

y = x – 4y = –3x

You should check that (1, –3) is actually a solution to both equations.