module

Practice Problem 6 Which set of equations has a solution of (3,1)?

line b and line d because they intersect each other at the point 3,1

Practice problem 9 y = 3x - 2 -6x + 2y = -4

the system has infinitely many solution

Practice problem 8 y = 3x + 4 y = 3x + 8

this system has no solution

How to graph linear equations using slope-intercept form. Equations must be written in slope-intercept form, or ___________________, where ____ represents the __________ and ____ represents the ____________________. Start by plotting the ___________________. Then use the _________ and follow ______ over ______ to plot the rest of the points.

y=m x+b m slope, b y-intercept ,rise run

Practice Problem 5 Ava's school took a field trip. A total of 32 vehicles were needed for the trip. Some students took the bus, and some students car-pooled. There were 27 people on each bus and 3 people in each car. 408 people altogether attended the trip. How many buses and cars were needed for the trip?

b number of buses c number of cars b+c=32 27b+3c=408 b=13 c=19

Types of solutions A system of linear equations might have ________ solution, ________ solution, or _______________ _________ solutions. Your friend is solving an equation and her last line of work says "5 = 5". What does this mean? Your friend is solving an equation and her last line of work says "1 = -2". What does this mean?

one, no,infinitely many

The solution to a system of linear equations is the _________________ __________ that make all of the equations in the system ______.

ordered pairs true

Is this a solution? To check the solution to a system, ___________ the values for each variable into each equation. If both equations are __________, then it is a solution.

plug,true

What is a solution? Solutions to linear equations include any _____________ located on the graphed line.

points

Practice Problem 5 y = -2x - 4 y = 4x + 2 y = -2x - 4 y = 4x + 2 Slope: _________ Slope: _________ Y-intercept: ____ Y-intercept: ____ Solution: _____________________

2,1 , -4 , 4/1 , 2 (-1/-2)

Practice Problem 4 y = 2x - 5 y = -23x + 3 y = 2x - 5 y = -23x + 3 Slope: ______ Slope: _________ Y-intercept: ____ Y-intercept: ____ Solution: _____________________ CHECK:

2/1 , -2/3 , -5 , 3 , 3,1

Practice Problem 4 Jacob bought some tickets to see his favorite singer. He bought some adult tickets and some children's tickets, for a total of 9 tickets. The adult tickets cost $10 per ticket, and the children's tickets cost $8 per ticket. If he spent a total of $76, then how many adult and children's tickets did he buy?

a number of adult ticket c number of child ticket a+c=9 10a+8c=76 c=7 a=2

Practice Problem 2 A movie theater charges $5 for an adult's ticket and $2 for a child's ticket. On Saturday, the theater sold 785 tickets for $3280. How many of each type of ticket were sold?

a number of adult's tickets c number of child's tickets a+c=785 5a+2c=3280 a=570 c=215

Practice Problem 6 Liam bought some notebooks and book covers for the upcoming school year. He bought a total of 12 items. Each notebook costs $1.33, and each book cover costs $1.10. He spent a total of $14.81. How many notebooks and how many book covers did he buy?

b number of the book cover c number of notebook b+n=12 1.10b+1.33n=14.81 n=7 b=5

Applying Systems of Equations Steps to Problem Solve Define your two ___________________. Be sure to write them out. Write your system of ___________________. _____________ the system of equations. Find the ____________ to the question being asked.

c=cost of an order chips t=cost of a taco 1c+2t=9.25 2c+5t=21.75 t3.25 c2.75

What is the elimination method? Elimination: Combining two or more equations in a system to _______________ (cancel) one ____________ to solve for the other; also called the _________________ ___________________.

eliminate, variable , addition ,method

Graphing is Estimation Graphing should be considered a method of _____________ and a useful tool for checking your work. The only way you can find an ____________ solution from the graph is if you draw a very neat coordinate plane system; if you draw very neat lines; if the solution happens to be a point with nice, neat integer coordinates; and if the lines are not close to being parallel. There is a lot of room for ____________ with graphing!

estimation,exact,error

Practice Problem 3 The drama club sells hot chocolate and coffee at the high school football games. At the last game, they had sales of $300. Monica knows that they used 275 cups that night. If the hot chocolate sells for $2 and coffee sells for $1, how much of each type of drink were sold?

h number of cups hot chocolate sold c number of cups coffee sold h+c=275 2h+c=300 h=25 c=250

One Solution A system has one solution if the lines ________________ at one point. Graph Equation: y = 3 x + 2 y= 2 x +5 Different ___________.

intersect,slopes

Graphing a System using a table Step 1: _____________ the y in each equation. y = 2x + 4 y = 4x + 2 Step 2: Create a __________ of values for each equation. Step 3: ____________ the points and connect then to form a ___________. Step 4: ___________ the solution Step 5: ________ your work

isolate substitution values for x to find y graph the point you found

Elimination Method part 2 -x + 2y = -13 2x + 3y = 12 In situations where neither of the variables in the two equations has opposite coefficients, it may be necessary to ___________________ one equations by a constant in order to create opposite terms. When multiplying an equation by a constant, remember to multiply _________ term within the equation!

multiply , all

No Solution A system will have no solutions if the lines ______________ intersect. Graph Equation:y = 3 x + 2 y = 3 x - 2 Same ________, but different ______________.

never,slopes,y-intercepts

Practice Problem 1 Graph y = 12 x + 6

next use the slope rise run to graph the next sets of points

Practice Problem 1 The talent show committee sold a total of 530 tickets in advance. Student tickets cost $3 each and the adult tickets cost $4 each. If the total receipts were $1740, how many of each type of ticket were sold?

s number of student tickets a number of adult tickets s+a=530 3s+4a=1740 a=150 s=380

Practice Problem 3 How many solutions does this system have? x + y = 3 3 x + 3 y = 9

solve each equation for y (y=m x+b) compare slopes and y intercept

Substitution Method __________________ one of the equations for "y =", if necessary. __________________ the value for "y" in the second equation. __________________ the new equation to find "x". __________________ the value for "x" in either of the original equations to find "y". __________________ your answer! Substitute x and y in both of the original equations to make sure they make the equations true.

solve,substitute solve, substitute ,check

Practice Problems 2 Is (2,2) the solution to this system? Explain your answer. Line A: -6 x + 2 y = -8 Line B: y = -2 x + 6

substitute the value for x and y into each equation simplify each equation

What is a system? A system of linear equations is two or more linear ________________ that use the same variables.

system of equation

Elimination Method In cases where the variables in two equations have opposite coefficients, we can solve the system by elimination with the following steps: Line up the like ___________ in the equations. _________ the equations to _____________ one variable and solve for the other. ________________ your answer into one of the original equations to solve for the other variable. Write your variables as an ____________ pair and _________ your answer. Practice Problem 1

terms , add , eliminate , substitute , ordered , check

Practice Problem 4 Which of the following equations would make a system that has a solution of (2,1)? y = 3 x - 4 2 x - 4 y = 0 x + 3 y = 7 5 x + 3 y = 13

test the point 2,1 in each equation

Practice Problem 7 How many solutions does each system have? Explain.

the first is no solution and the second one is infinite solution

Practice problem 7 -3x - 4y = 2 6x + 6y = -6

the solution for this system is (-2,1)

Practice problem 2 y = 9x - 2 y = 3x + 10

the solution for this system is (2,16)

Practice problem 3 y = 2x + 3 y = -2x + 15

the solution for this system is (3,9)

Practice problem 6 - 2x + y = -3 -6x + 4y = 4

the solution for this system is (8 ,13)

Practice Problem 1 y = 3x x + y = -32

the solution for this system is(-8,-24)

Practice problem 5 -2x - 3y = -7 y = 5x - 9

the solution for this system is(2,1)

Practice problem 4 y = x + 2 5x - 4y = -3

the solution for this system is(5 ,7 )

Practice Problem 1 6x + 2y = 4 -6x - 5y = -28

the solution to this system is (-2,8)

Practice Problem 3 5x + 2y = 18 -5x + 5y = 45

the solution to this system is (0,9)

Practice Problem 2 4x - 7y = 47 6x + 7y = -17th

the solution to this system is (3,-5)

Practice Problem 6 6x - 3y = -3 -5x + 6y = 41

the solution to this system is (5,11)

Practice Problem 5 y = 3x - 8 y = x + 4

the solution to this system is (6,10)

Practice Problem 4 4x - y = -19 -2x - 5y = -29

the solution to this system is(-3,7)

How many solutions does this system have? What is the solution?

there is only 1 solution because the lines intersect at one point

Infinitely Many Solutions A system has infinitely many solutions if the lines are on top of each other and always _____________. Graph Equations: y = 3 x+2-3 x + y = 2 Same ___________ and same _____________.

touching,slopes,intercept

An ordered pair is a solution to an equation if when its values are substituted in the equation and it makes the equation ___________.

true

Example: Is (2,4) a solution for the following system? y = 1/2 x + 3 y=2x

x,y

Practice Problem 7 The sum of two unknown numbers is 109 and the difference between the two numbers is 35. What is the value of the two unknown numbers?

x= first unknown number y= second unknown number x+y=109 x-y=35 x=72 y=37

Practice Problem 8 Sally needs to build a dog house. Woof construction charges $100 for the materials and $20 per hour to build the dog house. Doggy Condo Construction charges $50 for the materials and $30 per hour to build the dog house. How many hours will each company have to work for their cost to be equal? What will it cost?

x=number of hosts y= total cost y=20x+100 y=30x+50 y=20x5=100

Ordered pairs Written: (__y__, ___x__) Graphed: Start at the ______________. x: move _________ or __________ y : move ___________ or ________ (3,2) right 3 up 2 (4, -3)right 4 down 3 (-2, -5)left 2 down 5 (-6,6)left 6 up 6

y,x,origin,left,right,up,down

Practice Problem # 3 y = x + 1y = 2x - 4

y=01 y=11 y=21

Practice Problem # 2 y = 3x - 2y = -2x + 8 CHECK:

y=30-2 y=31-2 y=32-2 y=-2-2=8 y=-20+8 y=-22+8

Graphing using slope-intercept form Step 1: Isolate y in both equations Equations must be written in slope-intercept form, or ___________________, where ____ represents the __________ and ____ represents the ____________________. Step 2: Identify the slope and y-intercept for each equation Step 3: Graph each equation. Start by plotting the ___________________. Then use the _________ and follow ______ over ______ to plot the rest of the points.

y=m x+b, m,slope,b,y intercept y intercept rise run intercept check

Practice Problem 5 Create an equation that will make this system have infinitely many solutions. 4 x + 2 y = 6

you can create an equivalent equation by multiplying or dividing each term by the same factor