Unit 6 – Systems of Equations

Topic: Solving Systems of Linear Equations Algebraically

Substitution MethodIsolate a variable in one of the equations (doesn’t matter

which equation or which variable).Substitute the result into the other equation & solve for the

variable in question.Use that solution to substitute and solve for the other

variable.Elimination (Linear Combination) Method

Combine the two equations by adding or subtracting them, creating a single equation in which one variable has been eliminated, making it easy to solve for the other variable.Like terms & “=“ signs must be aligned.Each equation should have the same coefficient in front of one

of the variables.Use that solution to substitute and solve for the other

variable.

Solving Linear Systems Algebraically

5

153

873

x

x

x

8)7(2

82

xx

yx

The variable y is already isolated in the 2nd equation. Substitute the right side of that equation for y in the 1st equation; always protect your substitution with ( ).

Now solve this equation for x.

Substitute this value into one of the original equations and solve for y; it doesn’t matter which equation, but to keep things simple we’ll use the 2nd one.

2

75

7

y

y

xy

The solution to the system is the point (5, -2).

7

82

xy

yx{Solving Linear Systems: Substitution

00

033

yy

03)3(

03

yy

xy

The variable x is already isolated in the 2nd equation. Substitute the right side of that equation for x in the 1st equation; remember to protect your substitution with ( ).

Now solve this equation for y.

Our variable disappeared and left us with a true statement, meaning this equation is an identity and is always true. Therefore, our system is dependent and has infinite solutions.

3

03

yx

xy{Solving Linear Systems: Substitution

45 5

)192(

263

x

yx

yx

9

455

x

x

Each y term has the same coefficient. Since one is positive and one is negative, we can add the two equations together to eliminate y.

Now solve this equation for x.

Substitute this value into one of the original equations and solve for y. Since y is positive in the 1st equation, that’s probably the easier one to simplify.

1

2627

26)9(3

y

y

y

Solving Linear Systems: Elimination

192

263

yx

yx{

The solution to the system is the point (-9, 1).

102

2)2042(

yx

yx

Neither equation has the same coefficient in front of one of the variables. However, we can simplify the 2nd equation by a factor of 2, which would give both x terms the same coefficient.

Solving Linear Systems: Elimination

2042

173

yx

yx{

7

)102(

173

y

yx

yx

4

1721

17)7(3

x

x

x

Now we can combine the equations to eliminate x. Since both coefficients are the same sign, we must subtract (WATCH YOUR SIGNS!)

That was easy! Now we substitute this into one of the original equations and solve for x.

Solving Linear Systems: Elimination

102

173

yx

yx{

The solution to the system is the point (-4, 7).

How do I know what method to use?Graphing

Both equations are already in slope-intercept form, or can easily be rewritten in slope-intercept.

SubstitutionOne of the equations already has a variable isolated,

or a variable can be easily isolated.Elimination

One of the variables has the same coefficient in both equations, or one equation can be easily divided or multiplied to get the same coefficient.

REMEMBER: Each method is equally valid. The goal is to select the most efficient method depending on the system given.

JOURNAL ENTRYTITLE: Checking My Understanding: Solving

Systems of Linear Equations AlgebraicallyReview your notes from this presentation & create

and complete the following subheadings in your journal:“Things I already knew:” Identify any information

with which you were already familiar.“New things I learned:” Identify any new information

that you now understand.“Questions I still have:” What do you still want to

know or do not fully understand?

HomeworkQuest: Solving Systems of Equations

AlgebraicallyDue 1/23 (A-day) or 1/24 (B-day)