# Solving Linear Systems

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```					Solving Linear Systems
Using Substitution, Elimination,
and Graphing to solve a system of
equations
Review Graphing A Line
Define a Linear System
Slope Intercept Form
Standard Form of a Line
Solving a Linear System
Parallel Lines
Perpendicular Lines
Choose between Substitution and Elimination
Solving a System using Substitution
Solving a System using Elimination
Review of Graphing a Line
Put equation into slope intercept form
   Practice – Solve for y: 4x - 3y = -9
Identify the slope and the y-intercept
Locate the y-intercept on the y-axis
Starting at that point, move up (+) or
down(-), then right
   Show me how
Continue
Solve for y: 4x - 3y = -9
Step 1: Subtract the X term
   - 3y = - 4x – 9
Step 2: Divide by the coefficient of y
   Do NOT divide by y, JUST the coefficient
   -3y = -4x -9
-3   -3 -3
End result:       y = 4x + 3
3
Back
Graph a line example
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Continue
Graphing Practice
Graph the following equations:
   5x – 6y = 8

   2x + 4y = -6

   y = -3 x + 2
2
Define a Linear System
2 linear equations create a linear system

Can be in slope intercept or standard
form
Y form (Slope Intercept Form)
Slope intercept form y=mx+b

m is the slope (steepness of line)
b is the y-intercept (shows where the
line crosses the y axis)
Y intercept is also where x = 0 (0,b)
Standard Form of a Line
Ax + By = C
A, B, and C are integers
   NO FRACTIONS, NO DECIMALS
   A, B and C can be NEGATIVE
Solving a Linear System
What does this mean?
   Locating where the lines cross
   Where lines share a coordinate (x,y)

How many solutions?
   Parallel lines – no solutions
   Same equations – infinite solutions
   Any other lines – one solution
 Perpendicular lines
Parallel Lines
Same slope
Lines don’t cross – no solution
Solve equation – answer is two different
numbers
EXAMPLE: y=3x+4
y=3x-5
   If slope is the same and y-intercept isn’t,
the lines are parallel
Same Equations
Place both into slope intercept form
If the equations are identical
   Lines are located on top of one another
   End result – ONE line
Perpendicular Lines
One solution – the lines cross
Cross at a 90 degree angle
How to tell if two lines are
perpendicular?
   Put both equations in slope intercept form
   Multiply the slopes
   If answer = -1, they are perpendicular
Choose between Substitution and
Linear Combinations (Elimination)

Solve with Substitution if:
   One equation solves for x (i.e. x=)
   One or both equations are in y form
(slope-int)
Solve with Linear Combinations
(Elimination) if:
   Both equations are in Standard Form with
coefficients other than 0
Steps for Solving a System
with Substitution
Replace (substitute) one variable, and
solve for the other (i.e. replace y and
solve for x, or replace x and solve for y)
Replace that variable in one equation
and solve for the other variable

Continue
Solving a System with Substitution
Part 1
EXAMPLE:
   2x-3y=5
   y=-2x+1

replace y in first equation with -2x+1
2x - 3 (-2x + 1) = 5

Continue
Solving a System with Substitution
Part 2
Distribute and combine like terms

2x - 3 (-2x + 1) = 5
2x + 6x - 3     =5
8x-3         =5
Solve for x
8x = 8
x=1
Continue
Solving a System with Substitution
Part 3
Solve for the second variable
Replace x into one equation
2(1) - 3 y = 5

Solve for y
2 - 3y = 5
- 3y = 3
y = -1
The solution is (1, -1)
Continue
Solving a System with Substitution
Part 4

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

y = -2 x +1
-1 = -2 (1) +1        Both answers check
-1 = -2        +1                 Continue

-1 = -1
Solve a System – Elimination
Part 1

Be sure both equations are in Standard
Form
3x + 5y = 8
-6x – 3y = -2
Solve a System – Elimination
Part 2

Multiply one or both equations by a
constant
END RESULT – coefficients of one
variable are + and – of same number
2 (3x + 5y = 8)         6x + 10y = 16
-6x – 3y = -2        -6x – 3y = - 2
Solve an System – Elimination
Part 3
Combine the equations
6x + 10y = 16
-6x – 3y = - 2
7y = 14

Divide by coefficient
y = 14/7
y=2
Solve an System – Elimination
Part 4
Solve for the second variable
Replace y into one ORIGINAL equation
3x + 5 (2) = 8

Solve for y
3x + 10 = 8
3x       = -2
x = -2/3
The solution is (-2/3, 2)
Solve an System – Elimination
Part 5
Replace x and y into both equations
3 x + 5 y =8             -6 x – 3 y = -2
3(-2/3) + 5 (2) = 8       -6(-2/3)-3(2) = -2
-2 + 10 = 8                 4 - 6 = -2
8 = 8                      -2 = -2

Both solutions check   (-2/3, 2)

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