Algebra I

Chapter 6 Notes

Section 6-1: Graphing Systems of linear equations, Day 1

What is a system of linear equations?

Consistent –

Inconsistent –

Independent –

Dependent –

Section 6-1: Graphing Systems of linear equations, Day 1

What is a system of linear equations?Two equations with two variables.Consistent – a system with at least one solution

Inconsistent – a system with no solutions

Independent – a system with exactly one solution

Dependent – a system with an infinite number of solutions

Section 6-1: Graphing Systems, Day 1

Number of Solutions

Exactly One Infinite No Solution

Terminology

Graph

Section 6-1: Graphing Systems, Day 1Steps for solving systems by graphing

Graph the system (Use a Ruler!)y = -3x + 10y = x – 21. Graph the first equation on the graph2. Graph the second equation on the graph3. Find where the lines intersect4. CHECK your answer

Section 6-1: Graphing Systems, Day 1

Solve the following systems by graphingEx) y = ½ x Ex) 8x – 4y = 16

y = x + 2 -5x – 5y = 5

Section 6-1 Graphing Systems, Day 2

Systems that have no solutions –

Systems that have an infinite number of solutions -

Section 6-1 Graphing Systems, Day 2

Systems that have no solutions – Lines that are parallel and therefore never intersect

Systems that have an infinite number of solutions – Equations that end up graphing the same line

Section 6-1 Graphing Systems, Day 2

Solve the following systems by graphingEx) 2x – y = -1 Ex) y = -2x - 3

4x – 2y = 6 6x + 3y = -9

Section 6-1 Graphing Systems, Day 2

Use the graph to determine whether each system is consistent or inconsistent, independent or dependent.Ex) y = -2x + 3

y = x – 5

Ex) y = -2x – 5 y = -2x + 3

Section 6-2: Solving Systems by Substitution, Day 1

Steps for solving using substitution:

1) Solve ONE equation for ONE variable (Choose the a variable with a coefficient of 1 or -1 to make it easy)2) Substitute the expression from step 1 into the OTHER equation for the variable3) Solve the new equation4) Plug in the solution from step 3 into either equation to find the other variable5) Check your answer!

Ex) y = 2x + 1 3x + y = -9

Section 6-2: Solving Systems by Substitution, Day 1

Solve the systems using substitutionEx) y = x + 5 Ex) x + 2y = 6

3x + y = 25 3x – 4y = 28

Section 6-2: Solving Systems by Substitution, Day 2

Special Case SolutionsSolve the systems using substitutionEx) y = 2x – 4 Ex) 2x – y = 8 -6x + 3y = -12 -2x + y = -3

Section 6-3: Solving systems using the elimination method (add/sub)

Steps for solving using the elimination method

1) Write the system so like terms are aligned2) Add or subtract the equations, elimination a variable and solve3) Plug in the solution from step 2 to find the other variable4) Check your answer!

Ex) 4x + 6y = 32 3x – 6y = 3

Section 6-3: Solving systems using the elimination method (add/sub)

Solve using eliminationEx) 4y + 3x = 22 Ex) 7x + 3y = -6

3x – 4y = 14 7x – 2y = -31

Section 6-4: Elimination with Multiplication, Day 1

Steps for solving using the elimination method

1) Write the system so like terms are aligned2) Multiply one or both equations by a number, or 2 different numbers to get like coefficients for one variable3) Add or subtract the equations, elimination a variable and solve4) Plug in the solution from step 2 to find the other variable5) Check your answer!

Ex) 5x + 6y = -8 2x + 3y = -5

Section 6-4: Elimination with Multiplication, Day 1

Solve using the elimination methodEx) 4x + 2y = 8 Ex) 6x + 2y = 2

3x + 3y = 9 4x + 3y = 8

Section 6-4: Solve using elimination, Day 2

Solve using elimination. Be careful of special cases.Ex) 3x + y = 5 Ex) x + 2y = 6 6x = 10-2y 3x + 6y = 8

Section 6-4: Solve using elimination, Day 2

Solve the following systems using eliminationEx) 8x + 3y = 4 Ex) 12x – 3y = -3 Ex) 8x + 3y = -7

-7x + 5y = -34 6x + y = 17x + 2y = -3

Section 6-5: Which method is best?

Method When to use it…

Graphing

Substitution

Elimination

Section 6-5: Best Method

Determine which method is best, then solve the system using that methodEx) 2x + 3y = -11 Ex) 3x + 4y = 11

-8x – 5y = 9 y = -2x - 1

Section 6-5: Word ProblemsEx) Jenny has $24 to spend on tickets at the carnival. The small rides cost $2 per ticket, and the large rides cost $3 per ticket. She buys a total of 7 tickets. How many small ride tickets did she buy? How many large ride tickets did she buy? Write and solve a system.

Ex) Martha has a total of 40 DVDs of movies and TV shows. The number of movies is 4 less than 3 times the number of TV shows. Write and solve a system to find the numbers of movies and TV shows she owned.

Section 6-6: Systems of Linear Inequalities

Steps for Solving Systems of Linear Inequalities

1) Graph the first equation• Choose the correct

line (Solid or dashed)• Shade the correct side2) Graph the second equation• Choose the correct line

(Solid or dashed)• Shade the correct side3) Darken the shaded areas that overlap

Ex) y > -2x + 1 y < x + 3

Section 6-6: Systems of Linear Inequalities

Solve the following S.o.L.E by graphingEx) x > 4 Ex) y > -2

y < x – 3 y < x + 9

Section 6-6: Graphing systems of linear inequalities

Solve the following S.o.L.E by graphingEx) 3x – y > 2 Ex) y > 3

3x – y < -5y < 1