Systems of EquationsSystems of Equations

Graphing Systems of Equations is Graphing Systems of Equations is used by people with careers in used by people with careers in

biological science, such as biological science, such as ecologists. ecologists.

An ecologist uses graphs of systems to study populations of organisms and how they relate to their environment.

VocabularyVocabulary

• The equations The equations x + y = 24x + y = 24 and and 3x + y = 3x + y = 4444 together are called a together are called a System of System of EquationsEquations..

• One method for solving a system of One method for solving a system of equations is to carefully equations is to carefully graph the equations graph the equations on the same coordinate planeon the same coordinate plane. The . The coordinates of the point at which the graphs coordinates of the point at which the graphs intersectintersect is the solution of the system. is the solution of the system.

A system of two linear equations has exactly A system of two linear equations has exactly one ordered pair as its solution when the one ordered pair as its solution when the graphs of the equations intersect at exactly graphs of the equations intersect at exactly one point.one point.

(-2,2)

What if…What if…

• the two linear equations are parallel ?the two linear equations are parallel ?

• (there will be no solutions)(there will be no solutions)

• the two linear equations are the same the two linear equations are the same line?line?

• (there will be infinitely many points)(there will be infinitely many points)

Another method is called Another method is called SubstitutionSubstitution

• Let’s do a problem.Let’s do a problem.• If 3 bags of apples and 2 bags of If 3 bags of apples and 2 bags of

oranges cost $3.10 and 2 bags of oranges cost $3.10 and 2 bags of apples and 4 bags of oranges cost apples and 4 bags of oranges cost $4.20, how much does each bag of $4.20, how much does each bag of apples and each bag of oranges cost?apples and each bag of oranges cost?

• Bag of apples = xBag of apples = x• Bag of oranges = yBag of oranges = y

If If 3 bags of apples3 bags of apples and and 2 bags of oranges2 bags of oranges cost cost $3.10$3.10 and and 2 2 bags of applesbags of apples and and 4 bags of oranges4 bags of oranges cost cost $4.20$4.20, how much , how much does each bag of apples and each bag of oranges cost?does each bag of apples and each bag of oranges cost?

• 3x + 2y = 3.13x + 2y = 3.1 2x + 4y = 4.22x + 4y = 4.2• Solve one equation for one variable.Solve one equation for one variable.• 3x + 2y = 3.13x + 2y = 3.1• -3x -3x-3x -3x• 2y2y = = -3x-3x + + 3.13.1• 2 2 22 2 2• y = -1.5x + 1.55y = -1.5x + 1.55

If If 3 bags of apples3 bags of apples and and 2 bags of oranges2 bags of oranges cost cost $3.10$3.10 and and 2 2 bags of applesbags of apples and and 4 bags of oranges4 bags of oranges cost cost $4.20$4.20, how much , how much does each bag of apples and each bag of oranges cost?does each bag of apples and each bag of oranges cost?y = -1.5x + 1.55y = -1.5x + 1.55

• 3x + 2y = 3.1 3x + 2y = 3.1 2x + 4y = 4.2 2x + 4y = 4.2

• 2x + 4( y ) = 4.22x + 4( y ) = 4.2• 2x + 4(-1.5x + 1.55) = 4.22x + 4(-1.5x + 1.55) = 4.2• 2x + -6x + 6.20 = 4.22x + -6x + 6.20 = 4.2• -4x + 6.20 = 4.2-4x + 6.20 = 4.2• - 6.20 -6.20- 6.20 -6.20• -4x-4x = = -2-2• -4 -4-4 -4• x = .5x = .5•

If If 3 bags of apples3 bags of apples and and 2 bags of oranges2 bags of oranges cost cost $3.10$3.10 and and 2 bags of apples2 bags of apples and and 4 bags of oranges4 bags of oranges cost cost $4.20$4.20, how much does each bag of apples and each , how much does each bag of apples and each bag of oranges cost?bag of oranges cost?X = .5X = .5

• 3x + 2y = 3.1 3x + 2y = 3.1 2x + 4y = 4.2 2x + 4y = 4.2• 3(.5) + 2y = 3.13(.5) + 2y = 3.1• 1.5 + 2y = 3.11.5 + 2y = 3.1• -1.5 -1.5-1.5 -1.5• 2y2y = = 1.61.6• 2 22 2• y = .8y = .8

If If 3 bags of apples3 bags of apples and and 2 bags of oranges2 bags of oranges cost cost $3.10$3.10 and and 2 bags of apples2 bags of apples and and 4 bags of oranges4 bags of oranges cost cost $4.20$4.20, how much does each bag of apples and each , how much does each bag of apples and each bag of oranges cost?bag of oranges cost?

• So we found out that So we found out that

• x = .5 or $.50 for a bag of applesx = .5 or $.50 for a bag of apples

• and y = .8 or $.80 for a bag of and y = .8 or $.80 for a bag of oranges.oranges.

Here’s the graph!Here’s the graph!

(.5

( .5, .8 )

3x+2y=3.40

y=-1.5x + 1.55

2x+ 4y = 4.20

y=-.5x + 1.05

Quick ReviewQuick Review

• When two linear equations intersect you When two linear equations intersect you have an ordered pair.have an ordered pair.

• This ordered pair is the solution for the This ordered pair is the solution for the Systems of Equations.Systems of Equations.

• You can graph the two equations and see You can graph the two equations and see where they intersect.where they intersect.

• You can also use Substitution and find the You can also use Substitution and find the solution without graphing.solution without graphing.

• It’s always a good idea to check solutions It’s always a good idea to check solutions to make sure we didn’t make a mistake.to make sure we didn’t make a mistake.

• I hear there is another way to find I hear there is another way to find the solution to a system of the solution to a system of equations. equations.

• It’s called EliminationIt’s called Elimination

• But that’s another Power Point.But that’s another Power Point.