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NAME ______________________________________________ DATE______________________________ PERIOD _____________

3-1 Study Guide and InterventionSolving Systems of Equations

Solve Systems Graphically A system of equations is two or more equations with the same variables. You can solve asystem of linear equations by using a table or by graphing the equations on the same coordinate plane. If the lines

intersect, the solution is that intersection point. The following chart summarizes the possibilities for graphs of two linear 

equations in two variables.

Graphs of Equations Slopes of Lines Classification of System Number of Solutions

Lines intersect Different slopes Consistent and independent One

Lines coincide (same line) Same slope same y !intercept Consistent and dependent Infinitel" man"

Lines are parallel Same slope different y !intercepts Inconsistent None

Example: Graph the system of equations and describe it as  x – 3 y = 6consistent and independent , consistent and dependent , or inconsistent  ! x – y = –3

Write each equation in slopeintercept form.

 x ! " y # $ %  y #1

3  x ! &

& x ! y # !" %  y # & x ' "

The graphs intersect at (!", !"). *ince there is one

solution, the system is consistent and independent.

Exercises

Graph each system of equations and describe it as consistent and independent , consistent and dependent , or

inconsistent 

" " x ' y # !& ! x ' & y # + 3 & x ! " y # $ x ' & y # - " x ! -+ # !$ y  x ! $ y # "

# & x ! y # " $  x ' y # !& 6 " x ! y # &

 x' & y # & x '  y2

 # !-  x ' y # $

C#apter $  5 Glencoe Algebra

8/16/2019 4 - Solving Systems of Equations

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NAME ______________________________________________ DATE______________________________ PERIOD _____________

3-1 Study Guide and InterventionSolving Systems of Equations

Solve Systems %l&ebraically To solve a system of linear equations by substitution, first solve for one variable interms of the other in one of the equations. Then substitute this e/pression into the other equation and simplify. To solve a

system of linear equations by elimination, add or subtract the equations to eliminate one of the variables.

Example ": 'se substitution to solve the system of equations ! x – y = ( x ) 3 y = –6

*olve the first equation for y in terms of x.

& x ! y # 0 %irst e&'ation

 !  y # !& x ' 0 S'tract  x from ot# sides*

 y # & x ! 0 M'ltipl" ot# sides " +,*

*ubstitute the e/pression & x ! 0 for y into the second

equation and solve for x.

 x ' " y # !$ Second e&'ation

 x ' "(& x ! 0) # !$ S'stit'te  x + - for y *

 x ' $ x ! &1 # !$ Distri'ti.e Propert"

1 x ! &1 # !$ Simplif"*

1 x # &-  Add / to eac# side*

 x # " Di.ide eac# side " /*

 2ow, substitute the value " for x in either original

equation and solve for y.

& x ! y # 0 %irst e&'ation

&(") ! y # 0 Replace x 0it# $*

$ ! y # 0 Simplif"*

 !  y # " S'tract 1 from eac# side*

 y # !" M'ltipl" eac# side " +,*

The solution of the system is (", !").

Example !: 'se the elimination method to solve the system of equations3 x – ! y = #

$ x ) 3 y = –!$

3ultiply the first equation by " and the second equation

 by &. Then add the equations to eliminate the y variable.

" x ! & y # Multiply by 3.  0 x ! $ y # -&

+ x ' " y # !&+ Multiply by 2.  - x ' $ y # ! +

-0 x # !"4

 x # !&

5eplace x with !& and solve for y.

" x ! & y #

"(!&) ! & y #

 !$ ! & y #

 !& y # -

 y # !+

The solution is (!&, !+)

C#apter $  6 Glencoe Algebra

8/16/2019 4 - Solving Systems of Equations

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Exercises

Solve each system of equations

" " x ' y # 1 ! & x ' y # + 3 & x ' " y # !"

 x ' & y # -$ " x ! " y # "  x ' & y # &

# & x ! y # 1 $  x ! y # $ 6 + x ' & y # -&

$ x ! " y # - & x ! y

2 # !$ x ! & y # !-

* & x ' y # 4 + 1 x ' & y # !- ( " x ' 4 y # !$

" x '3

2 y  # -&  x ! " y # !-"  x ! y # 0