How many solution points does a line have? Think about: 2x + y = 5 or y = - 2x + 5.

How many solution How many solution points does a line points does a line

have?have?

Think about:Think about:

2x + y = 5 or y = - 2x + 2x + y = 5 or y = - 2x + 55

Now change the question: How Now change the question: How many solution points can many solution points can two lines two lines

shareshare in common?in common?

Answers:Answers:

Null set (if they are parallel)Null set (if they are parallel)–This will be called an This will be called an INCONSISTENT SYSTEMINCONSISTENT SYSTEM

One point (if they cross)One point (if they cross)–This will be called a This will be called a CONSISTENT SYSTEMCONSISTENT SYSTEM

Infinite Set or All Pts on the Line (if same Infinite Set or All Pts on the Line (if same

line is used twice)line is used twice)–This will be called a This will be called a DEPENDENT SYSTEMDEPENDENT SYSTEM (It is (It is

also consistent. Dependent supersedes also consistent. Dependent supersedes

consistent.)consistent.)

Solution: Null set (if they are Solution: Null set (if they are

parallel)parallel)–This will be called an This will be called an INCONSISTENT INCONSISTENT

SYSTEMSYSTEM

One point (if they cross)One point (if they cross)–This will be called a This will be called a CONSISTENT CONSISTENT

SYSTEMSYSTEM

Infinite Set or All Pts on the Line (if Infinite Set or All Pts on the Line (if

same line is used twice)same line is used twice)–This will be called a This will be called a DEPENDENT DEPENDENT

SYSTEMSYSTEM (It is also (It is also ConsistentConsistent, but the , but the

term term DependentDependent supersedes consistent.) supersedes consistent.)

4

2

-2

-4

4

2

-2

-4

4

2

-2

-4

The SOLUTION of a The SOLUTION of a SYSTEMSYSTEM is the is the

INTERSECTION SETINTERSECTION SET

What solution points do What solution points do they share in common?they share in common?

OROR

Where do they intersectWhere do they intersecton a graph?on a graph?

If you look at two If you look at two different equations on different equations on

the same graph, we call the same graph, we call this a this a SYSTEM OF SYSTEM OF

EQUATIONSEQUATIONSThink about:Think about:

2x y 5

3x 2y 4

y 2x 5

3y x 2

2

Method 1: Estimate the Method 1: Estimate the SOLUTION of a SOLUTION of a SYSTEMSYSTEM on a on a

graph. Where do they graph. Where do they intersect?intersect?

y 2x 5

3y x 2

2

5

4

3

2

1

2 4

g x = -2x+5 f x = 3

2 x-2

Solution to this system appears to

be:

{(2, 1)}

Example: Use trace on your graphing Example: Use trace on your graphing calculator to estimate the SOLUTION of calculator to estimate the SOLUTION of

the the SYSTEMSYSTEM below. Where do graphs below. Where do graphs intersect?intersect?

3

2

1

-1

-2 2 4

g x = -1

3 x+1

f x = 9-x2

2

1y x 1

3

y 9 x

Example: Use trace on your graphing Example: Use trace on your graphing calculator to estimate the SOLUTION of calculator to estimate the SOLUTION of

the the SYSTEMSYSTEM below. Where do graphs below. Where do graphs intersect?intersect?

Solution to this system appears to include TWO pts:

{ (-2.38, 1.79), (3, 0) }

3

2

1

-1

-2 2 4

B: (3.00, 0.00)

A: (-2.38, 1.79)g x = -

1

3 x+1

f x = 9-x2

2

1y x 1

3

y 9 x

Summary of Method 1: Summary of Method 1: Estimate the SOLUTION of a Estimate the SOLUTION of a SYSTEMSYSTEM on a graph. on a graph. (Goal: (Goal:

Find intersection pts.)Find intersection pts.)

DisadvantagesDisadvantages: : Might give rough estimate onlyMight give rough estimate only

We might not know how to graph some We might not know how to graph some higher power equations yet.higher power equations yet.

AdvantagesAdvantages::When graphs are easy to sketch this is a When graphs are easy to sketch this is a good method to choose as a 2nd check!good method to choose as a 2nd check!

Method 2: Substitution Method 2: Substitution MethodMethod

(Goal: replace one variable(Goal: replace one variablewith an equal expression.)with an equal expression.)

Step 1: Look for a variable with a coefficient of one.

Step 2: Isolate that variable

Equation A now becomes: y = 3x + 1

Step 3: SUBSTITUTE this expression into that variable in Equation B

Equation B now becomes 7x – 2( 3x + 1 ) = - 4

Step 4:Solve for the remaining variable

Step 5:Back-substitute this coordinate into Step 2 to find the other coordinate. (Or plug into any equation but step 2 is easiest!)

3x y 1

7x 2y 4

Method 2: Method 2: Substitution Substitution (Goal: replace one variable(Goal: replace one variablewith an equal expression.)with an equal expression.)



Equation A now becomes:

y = 3x + 1


Equation B becomes

7x – 2( 3x + 1 ) = - 4

Step 4: Solve for the remaining variable

Step 5: Back-substitute this coordinate into Step 2 to find the other coordinate. (Or plug into any equation but step 2 is easiest!)

3x y 1

7x 2y 4

y

7x 2( ) 4

7x 6x 2 4

x 2

3x 1

3x 1

Arewedone

4

x 2 ?

Method 2: Method 2: Substitution Substitution (Goal: replace one variable(Goal: replace one variablewith an equal expression.)with an equal expression.)



Equation A now becomes:

y = 3x + 1


Equation B becomes

7x – 2( 3x + 1 ) = - 4



3x y 1

7x 2y 4

y

7x 2( ) 4

7x 6x 2 4

x 2 4

x 2 y

3x 1

3x 1

Solution : {(

3( 2) 1

y 5

2, 5)}

Example: Example: Substitution Substitution (Goal: replace one variable(Goal: replace one variablewith an equal expression.)with an equal expression.)






x 5y 8

2x 3y 3

A : x 5y 8 x

B : 2( ) 3y 3

10y 16 3y 3

13y 16 3

13y 13

y 1 x 5

5y 8

5y 8

y 8

Example: Example: Substitution Substitution (Goal: replace one variable(Goal: replace one variablewith an equal expression.)with an equal expression.)






x 5y 8

2x 3y 3

A : x 5y 8 x

B : 2( ) 3y 3

10y 16 3y 3

13y 16 3

13y

5y 8

5y 8

Solut

1

ion

3

y

:{(3,1)

1 x 5y 8

x 5(1) 3

}

8

What did you

just find?

Where dothe two linesintersect?

Method 2 Summary: Method 2 Summary: Substitution MethodSubstitution Method

(Goal: replace one variable with an equal (Goal: replace one variable with an equal expression.)expression.)

DisadvantagesDisadvantages: : Avoid this method when it requires messy Avoid this method when it requires messy

fractions fractions Avoid IF no coefficient is 1. Avoid IF no coefficient is 1.

AdvantagesAdvantages::This is the algebra method to use when This is the algebra method to use when degrees of the equations are not equal.degrees of the equations are not equal.

Step 1: Look for the LCM of the coefficients on either x or y. (Opposite signs are recommended to avoid errors.)

Here: -3y and +2y could be turned into -6y and +6y

Step 2: Multiply each equation by the necessary factor.

Equation A now becomes: 10x – 6y = 10

Equation B now becomes: 9x + 6y = -48

Step 3: ADD the two equations if using opposite signs (if not, subtract)

Step 4:Solve for the remaining variable

Step 5:Back-substitute this coordinate into any equation to find the other coordinate. (Look for easiest coefficients to work with.)

Method 3: Elimination Method 3: Elimination MethodMethodor Addition/Subtraction or Addition/Subtraction MethodMethod(Goal: Combine equations (Goal: Combine equations to cancel out one variable.)to cancel out one variable.)

5x 3y 5

3x 2y 16


Here: -3y and + 2y could be turned into -6y and + 6y


A becomes: 10x – 6y = 10

B becomes: 9x + 6y = -32



Step 5: Back-substitute this coordinate into any equation to find the other coordinate. (Look for easiest coefficients to work with.)

Method 3: Elimination or Addition/Subtraction Method 3: Elimination or Addition/Subtraction MethodMethod(Goal: Combine equations (Goal: Combine equations to cancel out one variable.)to cancel out one variable.)

6y

6y

5x 3y 5

3x 2y 16

10x 10

9x 48

19x 38

x 2 Arewedone?

+






Method 3: Elimination or Addition/Subtraction Method 3: Elimination or Addition/Subtraction MethodMethod(Goal: Combine equations (Goal: Combine equations to cancel out one variable.)to cancel out one variable.)

+

5x 3y 5

3x 2y 16

10x 10

9x 48

19x 38

x 2 5 3y 5

5( ) 3y 5

10

6y

6y

x

2

Solution :{( 2, 5)

3y 5

3y 15

y 5

}

What did you

just find?

Where dothe two linesintersect?






Example: Elimination or Addition/Subtraction Example: Elimination or Addition/Subtraction MethodMethod(Goal: Combine equations (Goal: Combine equations to cancel out one variable.)to cancel out one variable.)

+

2x 3y 2

5x 7y 34






Example: Elimination or Addition/Subtraction Example: Elimination or Addition/Subtraction MethodMethod(Goal: Combine equations (Goal: Combine equations to cancel out one variable.)to cancel out one variable.)

+

2x 3y 2

5x 7y 34

10x 15y 10

10x 14y 68

29y 58

y 2 2x 3(2) 2

2x

Solution :{(

6 2

2x

4,2)

8

}

x 4

Method 3 Summary: Method 3 Summary: Elimination MethodElimination Method

or Addition/Subtraction Methodor Addition/Subtraction Method(Goal: Combine equations to (Goal: Combine equations to

cancel out one variable.)cancel out one variable.)DisadvantagesDisadvantages: :

Avoid this method if degrees and/or formats of Avoid this method if degrees and/or formats of

the equations do not match.the equations do not match.

AdvantagesAdvantages::Similar to getting an LCD, so this is intuitive, Similar to getting an LCD, so this is intuitive, and uses only integers until the end of the and uses only integers until the end of the

problem.problem.

Which method would you choose?Which method would you choose?

Method 1: Graphing MethodMethod 1: Graphing Method

Method 2: Substitution MethodMethod 2: Substitution Method

Method 3: Elimination MethodMethod 3: Elimination Method

or Addition/Subtraction or Addition/Subtraction

MethodMethod

2

x 3y 11 8x 3y 11 x y 1A. B. C.

7x 4y 6 7x 4y 6 7x 4y 6

Example: WATCH FOR SPECIAL CASESExample: WATCH FOR SPECIAL CASES(What if both variables cancel?)(What if both variables cancel?)

+

Example: WATCH FOR SPECIAL CASESExample: WATCH FOR SPECIAL CASES(What if both variables cancel?)(What if both variables cancel?)

+

Looking for more info or practice?Looking for more info or practice?

Try these links:Try these links:

Cool Math ExplanationCool Math: Crunch Some Sample Problems

Hippocampus Lessons and Practice Problems

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How many solution points does a line have? Think about: 2x + y = 5 or y = - 2x + 5.

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