2 VARIABLE LINEAR SYSTEM

Mary Lam Teresa La

Kerry Huynh

LINEAR EQUATION

Standard Form of a linear equation:Ax+By=C

A, B and C are the integers of the equation Slope form of a linear equation:

y=mx+bm is the slopey and b are integers

HOW TO FIND SLOPE

Slope is the number multiplied by x. Slope is represented by m in the equation

y=mx+b Slope can be found by rise/run on a graph or

the equation y2-y1

x2-x1

FINDING SLOPE

Y=mx+b M is the slope of the equation.

Ex. Coordinates: (4,5)(6,4)

y2-y1 4-5 -1

x2-x1 6-4 = 2

Slope= -1 2

FINDING THE EQUATION

y=mx+b

coordinates: (4,5)

(6,4)

1. First find the slope, the slope is -1, it was found in the previous slide. Go back if you need help to find the slope. -1 replaces the m.

2. Choose and plug in a coordinate to find b. You can choose wither. Lets use (4,5), 5 replaces the y and 4 replaces the x.

5=-1(4)+b

3. Now solve for b

5=-1(4)+b distribute

5=-4+b add 4 on both sides

+4 +4

9=b the equation is y=-1x+9

FIND THE EQUATION

Example:Try this on your own, then check

the next slide for your answer. Coordinates: (3,7) and (5,12)

equation is y=mx+b

HOW TO FIND THE EQUATIONStep 1: Find your slope.

y2-y1 12-7 = slope is 5

x2-x1 5-3 = 2

Step 3: Choose and plug in a coordinate using the equation y=mx+b. You can use either coordinate. We will use (3,7). The 3 replaces the x and the 7 replaces the y, slope replaces the m.

7=(5/2)(3)+b

Step 4: solve to find b

7=(5/2)(3)+b distribute

7=7.5+b subtract 7.5 on both sides to get b

-7.5 -7.5

-.5=b

The equation is y= (5/2)x-0.5

HOW TO GRAPH

Y =1x+2 the 1 would be the slope which is how much

the line rises by. b would be the y intercept.

Y=1X+2

1 2 3 40

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

DEFINITION OF LINEAR SYSTEMS( 2 VARIABLES)

A linear system is a system of equation that involves the same set of variables.

Example of linear equation:2y+5x=126y+3x=24

SOLVING BY SUBSTITUTING

2y+4x=12y=x

SOLUTIONS

2y+4x=12y=x

Step 1: substitute y in for the equation 2y+4x=122(x)+4x=12

Step 2: Distribute2x+4x=12

Step 3: Combine like terms6x=12

Step 4: Solve for x by diving both sides by 66x=12 ÷6 ÷6X=2c

SOLVING BY ELIMINATION

2y+6x=12y+2x=10

HOW TO SOLVE-ELIMINATION2y+6x=12

y+2x=10

Step 1: Multiply one variable in an equation to equal the other equation

2y+6x=12

2(y+2x=10)

Step 2: Distribute the equation

2y+6x=12

2y+4x=20

Step 3: Subtract the top equation from the bottom equation.

2y+6x=12

2y+4x=20

2x=-8

÷2= ÷2 X=-4

SOLVING WITH WORD PROBLEMS #1

George bought 64 oranges (x) for his school festival along with 45 apple(y). The total price came to $63. If the price of 5 oranges was equal ton the price of 3 apples, how much would each orange cost?

HOW TO SOLVE Step1: Set up the equation

64x+45y=$635x=3y

Step2: Choose a method. In this case, substituting would be the best choice.

5x=3y3 35x =y (Solve for Y first before solving

for X)3

Step3: Substitute (y) in the equation 64x+45y=$63 to solve for oranges (x)

1. 64x+45(5x)=$63 3

2. 64x+75x = $63 3. 139x=$63

-139 -139 4. x= $0.45

Step4: Write down the answer in complete sentences.Each orange would cost a total of $0.45

SOLVING WITH WORD PROBLEMS #2 Frank sold mangos(x) and peaches(y) for a total of

500 boxes of fruits. The mango costs $4per box and the peaches costs $3 per box. The total profit he made was $1600. How many boxes of mangos and peaches did he sell?

HOW TO SOLVE

Step 1: Set up the equations.x+y=5004x+3y=1600

Step 2: Multiply one variable in an equation to equal the other equation.4(x+y)=5004x+3y= 1600

Step 3: Distribute and Solve.4x+4y=2000 4x+4y=20004x+3y=1600 - 4x+3y=1600

y=400Step 4: Solve for x using y in one of the equations.

x+(400)=500 - 400= -400

x= 100

There are 100 boxes of mangos and 400 boxes of peaches.

SOLVING WORD PROBLEMS WITH SLOPE INTERCEPT FORM EQUATIONS. (ELIMINATION)

Sally’s company makes a total of $375 for every Gold necklace she sells and a additional charge of $2 for every gift bags. She makes a total of $7520 at the end of one week. John’s company makes a total of $425 for every necklace he sells and a additional charge of $4 for every gift bags. He makes a total of $8540 at the end of one week.

A. Write two equations using the information above.

B. What is the total Necklace Sally and John both sold?

C. How much gift bags were sold from each?

D. Graph both equations and decide which company makes more money.

HOW TO SOLVE A) Step one, plug what is given into the formula C=ax+by

Sally: $7520= $375X + 2BJohn: $8540= $425X+4B

B) To find the total necklace that Sally and John sold, simply solve for one variable and substitute. Elimination would be the best choice. 2($7520= $375X + 2B) Multiply by 2 in order to cancel out one variable (B) $8540= $425X+4B $15040=$750X+4B Now subtract. - $8540= $425X+4B

-6500=$-325X Divide by both side to get X by itself.

-325 -325

C) To find the total number of gift bags both sold, use substitution. Plug in one of the equations and solve.

$7520=$375(20)+2B$7520=7500+2B

-7500 -7500 20=2B

2 2

20=X

10=B

D) GRAPHING THE SYSTEM. y

1400 1300 1200 1100 1000 900 800 700 600 500 400 300 200 100 0 1 2 3 4 5 6 7 8 9 10

John

Sally

Sally would be the best choice because the price is lower.

GRAPHING INEQUALITY SYSTEMS- DON’T FORGET TO SHADE IN THE REGIONS. QUADRANT 1 AND 4 SHOULD BE SHADED SO THAT IT INDICATES THE INTERSECTION BETWEEN 1 AND 2.

Y > -6x-2 Y<2x+4

1

2

SOLVING WORD PROBLEMS WITH SLOPE INTERCEPT FROM EQUATIONS.(SUBSTITUTING)

Joe and Sally are selling baked goods for a school fundraiser that helps raise money for charities around the world.

A. Using the points (2,30) and (3,35) find the slop of Joe’s equation.

B. Using the points (2,24) and (4,28) find the slope of sally’s equation.

C. Find the y-intercept of the 2 equations, if Joe made a total of $45 and Sally made a total of $30.

HOW TO SOLVE

A: Find slope for 1st equation:y1 – y2 35 – 30 = 5 (slope= 5) x1 – x2 3 – 2 1

B: Find slope for 2nd equation:28 – 24 = 4 (slope= 2)4 – 2 2

C: Setting up the equations: (y=mx+b) Joe: Sally:y= 5x + b y=2x+b 45= 5x +b 45=5(5) +b plug in the x.-30 = 2x+b 45= 25 +b15 = 3x -25 -25 3 3 20 = b solve for b. 5=x