Systems of Equations Substitution Method & Elimination Method copyright © 2011 by Lynda Aguirre1

Systems of Equations

Substitution Method& Elimination Method

copyright © 2011 by Lynda Aguirre 1

Substitution Method

Systems of Equations2 Equations in 2 variables



Substitution MethodSubstitution MethodThis method takes one equation and substitutes it into the other one.This method takes one equation and substitutes it into the other one.

Why are we doing this? To get an equation with only one variable (unknown value) in it.

Step 1: Solve either equation for x or y (this will be the “original” equation)Sometimes an equation

already has an equation that is solved for x or y.

Step 2: Replace x or y in the “other” equation with the value from the “original” equation.

Now we have an equation with only one

variable in it.Step 3: Solve for the remaining variable (in this case: x)

This gives us one variable (x=4), now we need to find

the other one (y).

Add Like Terms and Isolate x.

Call this equation the “original”


Substitution MethodSubstitution MethodPart II: Find the other variable

In steps 1-3, we plugged the 2nd equation into the

1st and found x=4.

Step 4: Plug the value from step 3 (x=4) into the “original” equation

Step 5: Solve for the remaining variable (y).This gives us both values

which we list as a coordinate

Solution: Solution: (4, 1)(4, 1)

3)4( y

1y


Substitution MethodSubstitution MethodStep 1: Solve either equation for x or y (your choice)

Step 2: Replace the variable in the “other” equation with the value from the “original” equation

My choice: Solve the 1st equation for y:

1st equation is the“original” equation

2nd equation (the “other” equation)

The variable

The value of y

Step 3: Solve for the remaining variable (in this case, solve for x)

This gives us one value(4, ___)

Now we need to find the “y”

Add Like Terms and Isolate x.

1st equation:

2nd equation:


Substitution MethodSubstitution MethodPart II: Find the other variable

Step 4: Plug the value from step 3, (x=4), into the “original” equation

Step 5: Solve for the remaining variable (y).

This gives us both values which we list as a

coordinate

Solution: Solution: (4, 2)(4, 2)

The original equation

copyright © 2011 by Lynda Aguirre7

STEPS

1) Solve one equation for x or y, label it “original”

2) Plug “original” into the “other” equation

3) Solve for 1st variable

4) Plug 1st variable into the “original” equation

5) Solve for 2nd variable6) Write the solution (x, y)

Note: if the problem has letters other than x and y in it,put them in alphabetical order

Solution: (6, 0)

Substitution MethodSubstitution MethodTry this one on your own


STEPS

1) Solve one equation for x or y, label it “original”

2) Plug “original” into the “other” equation

3) Solve for 1st variable

4) Plug 1st variable into the “original” equation

5) Solve for 2nd variable6) Write the solution (x, y)

Note: if the problem has letters other than x and y in it,put them in alphabetical order

Substitution Method: stepsSubstitution Method: steps

Things to note:

--In step 1, if you choose to solve for a variable with a coefficient, you will create fractions.

--You must substitute into one equation in step 2 and then the other one in step 4

--You can check your answers by plugging the numbers (x,y) into BOTH equations

--Sometimes step 1 is not necessary if one of the equations is already solved for x or y



Dependent and Inconsistent Systems of Dependent and Inconsistent Systems of EquationsEquationsAll the examples up to this point were systems of equations that (if

graphed) cross at a single point.

But it is possible for two lines to be parallel (i.e. they never cross)

A system of parallel Lines is called an A system of parallel Lines is called an “Inconsistent System”“Inconsistent System”

The same line graphed twice is called a The same line graphed twice is called a ““Dependent System”Dependent System”

Lines that cross at a point (x, y) are Lines that cross at a point (x, y) are “Consistent Systems”.“Consistent Systems”.

OR Two lines could represent the same line graphed twice (i.e. one on top of the other, so they intersect at every point)


Types of Systems and SolutionsTypes of Systems and Solutions

Type of System

Solution Graph

Consistent (x,y) Two lines that cross

Inconsistent No solution Parallel linesDependent An Infinite

Number of Solutions

Same line twice (looks like

one line)


Solution: no solutionTOS: Inconsistent


Solution: An infinite number of solutionsTOS: Dependent


Elimination Method

Systems of Equations2 Equations in 2 variables



Elimination (or Addition) MethodElimination (or Addition) MethodThis method takes one equation and adds it to the other one.This method takes one equation and adds it to the other one.

Why are we doing this? To get an equation with only one variable (unknown value) in it.

Sometimes one or both equations are already in the

correct format

Step 2: Draw a line underneath and add the like terms (straight down). One should cancel out.

Now we have an equation with only one

variable in it.

Step 3: Solve for the remaining variable (in this case: x)

Step 4: Substitute this value into “either” of the original equations

Solution:(4 , 2)


Elimination (or Addition) MethodElimination (or Addition) Method

Sometimes one or both equations are already in the

correct format

Step 2: Draw a line underneath and add the like terms (straight down). One should cancel out.

Now we have an equation with only one variable in it.

Step 3: Solve for the remaining variable (in this case: y)

Step 4: Substitute this value into “either” of the original equations Solution:( -4, 7)

Step 1: Put both equations into General Form

Step 1a: If necessary, multiply one equation (or both) by a number and/or a negative sign so x’s or y’s will cancel (i.e. equal zero)when added

My choice: Make the x’s cancel

Now the x’s have the same number and different signs


Elimination (or Addition) MethodElimination (or Addition) MethodSolution: (-4, 7)Try this one on your own

ELIMINATION STEPS

Preparation:

Step 1: Put both equations in General Form:

Step 1a: Multiply one (or both) equations by a constant if necessary

Elimination Process:

Step 2: Draw a line underneath and add the like terms (straight down), x’s or y’s should cancel

Step 3: Solve for the remaining variable

Step 4: Plug the value from step 3 into either of the original equations


Write the solution as a point: (x, y)


Elimination (or Addition) MethodElimination (or Addition) MethodSolution: (-4, 1)Try this one on your own

ELIMINATION STEPS

Preparation:










Elimination (or Addition) MethodElimination (or Addition) MethodSolution: (-1, -6)Try this one on your own

ELIMINATION STEPS

Preparation:










Elimination (or Addition) MethodElimination (or Addition) MethodSolution: (-8, -2)

ELIMINATION STEPS

Preparation:









Try this one on your own


Preparation:


Step 1a: Multiply one (or both) equations by a constant


Step 2: Draw a line underneath and add the like terms (straight down)





Elimination Method: stepsElimination Method: steps

Things to note:

--You can check your answers by plugging the numbers (x,y) into BOTH equations

--Sometimes step 1 is not necessary if the equations are already in General Form

--If there are fractions in either equation, multiply by the LCD to get rid of them

--If there are decimals in either equation, multiply by a power of 10 (10, 100, 1000,…)







Using the Elimination Method, name the Solution and the Type of System (TOS)Using the Elimination Method, name the Solution and the Type of System (TOS)

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Systems of Equations Substitution Method & Elimination Method copyright © 2011 by Lynda Aguirre1