Systems of Linear Systems of Linear Equations Equations

Math 0099Math 0099

Section 8.1-8.3Section 8.1-8.3

Created and Presented by Created and Presented by Laura Ralston Laura Ralston

What is a system of linear What is a system of linear equations?equations?

It is two or more linear equations with It is two or more linear equations with the same variables considered at the the same variables considered at the same time. same time.

Number of variables equals number of Number of variables equals number of linear equations in the systemlinear equations in the system

Examples: Examples: x + y = 4 x + y = 4 x + y = 2x + y = 2

x+ y = -10x+ y = -10 3x + 4y = 73x + 4y = 7

What is the solution set to What is the solution set to a system? a system?

The solution set to the system of linear equations is ALL ordered pairs that are solutions to both equations, that is, makes both equations TRUE at the same time.The solution set to the system of linear equations is ALL ordered pairs that are solutions to both equations, that is, makes both equations TRUE at the same time. To decide whether an ordered pair is a solution to a system, substitute the values for x and y in both equations. If the results for both equations are true then the ordered pair is a To decide whether an ordered pair is a solution to a system, substitute the values for x and y in both equations. If the results for both equations are true then the ordered pair is a

solution to the system. solution to the system.

Example: Determine if the given Example: Determine if the given point is a solution to the system. point is a solution to the system.

x + y = 2x + y = 2

3x + 4y = 73x + 4y = 7

(1, 1) (1, 1)

(4, -2) (4, -2)

Questions to AnswerQuestions to Answer

How do we find the solution, if How do we find the solution, if there is one? there is one?

Will there always be a solution to a Will there always be a solution to a system of linear equations?system of linear equations?

Can there be more than one Can there be more than one solution? solution?

Methods for Solving a Methods for Solving a System of EquationsSystem of Equations

* Graphing - Section 8.1Graphing - Section 8.1* Substitution - Section 8.2 Substitution - Section 8.2 * Addition (Elimination)Addition (Elimination)

Section 8.3Section 8.3

GRAPHING ProcedureGRAPHING Procedure

1.1. Graph the first equation in the Graph the first equation in the coordinate plane coordinate plane

2.2. Graph the second equation on the Graph the second equation on the same coordinate plane same coordinate plane

3.3. Record the coordinates of the point Record the coordinates of the point of intersection of the two graphs. of intersection of the two graphs. This ordered pair is the solution to This ordered pair is the solution to the systemthe system

4.4. Check solution in both equations. Check solution in both equations.

Three possibilities for Three possibilities for solutions for a system solutions for a system

NO SOLUTIONNO SOLUTION– Graphically, the Graphically, the

lines would be lines would be parallel. parallel.

– Solving for x will Solving for x will result in a false result in a false statement with no statement with no variable remaining variable remaining

– INCONSISTENTINCONSISTENT

ONE SOLUTIONONE SOLUTION– Graphically, the Graphically, the

lines will intersect lines will intersect ONCE. Solution ONCE. Solution will be an ordered will be an ordered pairpair

– Solving for x will Solving for x will result in a result in a numerical value numerical value

CONSISTENT CONSISTENT

INFINITE INFINITE SOLUTIONSSOLUTIONS– Graphically, the Graphically, the

lines coincide (same lines coincide (same line) line)

– Solving for x results Solving for x results in a true statement in a true statement with no variable with no variable remaining remaining

– DEPENDENTDEPENDENT

ExamplesExamples

x + 2y = 8x + 2y = 8 2x – y = 12x – y = 1

y = 2x + 5y = 2x + 5 4x – 2y = -104x – 2y = -10

AssignmentAssignment

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SUBSTITUTIONSUBSTITUTION

Objective is to eliminate one of the Objective is to eliminate one of the variables so that a new equation is formed variables so that a new equation is formed with just one variablewith just one variable

Most useful when one of the equation is Most useful when one of the equation is solved for one variable already OR if one solved for one variable already OR if one of the variables has a coefficient of 1; of the variables has a coefficient of 1; otherwise, you get Fractions !!! otherwise, you get Fractions !!! Fractions !!! Fractions !!!Fractions !!! Fractions !!!

Provides exact answers rather than Provides exact answers rather than estimations estimations

Substitution Steps Substitution Steps

1 Solve one of the given equations for Solve one of the given equations for either x or y, whichever is easier. either x or y, whichever is easier.

2 Substitute the result from step 1 into Substitute the result from step 1 into the other given equation the other given equation

3 Solve for the remaining variable Solve for the remaining variable 4 Substitute (“back substitute”) this Substitute (“back substitute”) this

solution into one of the ORIGINAL solution into one of the ORIGINAL given equations given equations

Substitution steps Substitution steps continued ...continued ...

5 Solve for the variable. Write final Solve for the variable. Write final solution as an ordered pair (x, y) solution as an ordered pair (x, y)

6 Check answer in both given Check answer in both given equations. True statements equations. True statements indicate correct answers. indicate correct answers.

ExamplesExamples

x + y =3x + y =3 y = -3 =2xy = -3 =2x y = 2x y = 2x 4x – 2y = 64x – 2y = 6

y = 4 – 3xy = 4 – 3x Y=-3x + 6Y=-3x + 6

AssignmentAssignment

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ADDITION (ELIMINATION)ADDITION (ELIMINATION)

The idea is to eliminate one of the The idea is to eliminate one of the variables from the system of linear variables from the system of linear equations. equations.

To do this, one of the variables To do this, one of the variables must have coefficients that are must have coefficients that are opposites. opposites.

Provides exact answers rather than Provides exact answers rather than estimated ones estimated ones

Addition (Elimination) Addition (Elimination) Steps Steps

1 Write each equation in standard form Write each equation in standard form (align like terms) (align like terms)

2 If needed, multiply one or both equations If needed, multiply one or both equations by appropriate number(s) so that the by appropriate number(s) so that the coefficients on either x or y are opposites. coefficients on either x or y are opposites.

3 Add the equations from step 2 together Add the equations from step 2 together by combining like terms. This should by combining like terms. This should result in an equation with one variable. result in an equation with one variable.

4 Solve the equation from step 3. Solve the equation from step 3.

Addition steps Addition steps continued…..continued…..

5 Back Substitute the solution from step 4 Back Substitute the solution from step 4 into either of the ORIGINAL given into either of the ORIGINAL given equation equation

6 Solve for the other variable. Write final Solve for the other variable. Write final answer in an ordered pair (x, y) answer in an ordered pair (x, y)

7 Check your answer in each original Check your answer in each original given equation. True statements result given equation. True statements result in correct answers. in correct answers.

ExamplesExamples

2x + 2y = 42x + 2y = 4 x – y = -3x – y = -3

y = 3x + 15y = 3x + 15 6x – 2y = -306x – 2y = -30

2x – 5y = 62x – 5y = 6 4x – 10y = -24x – 10y = -2

AssignmentAssignment

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COMPASS Practice COMPASS Practice QuestionsQuestions

What is the solution of the system of What is the solution of the system of equations below? equations below?

A. (3a, 2a) A. (3a, 2a)

B. (-3a, 2a) B. (-3a, 2a) 3x + 4y = a3x + 4y = a

C. (15a, 11a) C. (15a, 11a) 2x – 4y = 14a2x – 4y = 14a

D. (15a, -11a) D. (15a, -11a)

E. (3a, -2a) E. (3a, -2a)

What are the (x, y) coordinates of What are the (x, y) coordinates of the point of intersection of the the point of intersection of the lines determined by the equations lines determined by the equations 2x – 3y = 4 and y = x? 2x – 3y = 4 and y = x?

A. (4, 4) A. (4, 4) B. (–4, –4) B. (–4, –4) C. (–4, C. (–4, 4) D. (4, –4) 4) D. (4, –4) E. (2, 0) E. (2, 0)