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2 Variable Linear System

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+b m is the slope

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2 Variable Linear System

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  1. 2 Variable Linear System Mary Lam Teresa La Kerry Huynh

  2. 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+b m is the slope y and b are integers

  3. 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

  4. 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

  5. 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

  6. 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

  7. How to Find the Equation Step 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

  8. 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.

  9. Y=1x+2

  10. 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=12 6y+3x=24

  11. Solving by Substituting 2y+4x=12 y=x

  12. Solutions 2y+4x=12 y=x Step 1: substitute y in for the equation 2y+4x=12 2(x)+4x=12 Step 2: Distribute 2x+4x=12 Step 3: Combine like terms 6x=12 Step 4: Solve for x by diving both sides by 6 6x=12 ÷6 ÷6 X=2c

  13. Solving by Elimination 2y+6x=12 y+2x=10

  14. How to Solve-Elimination 2y+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

  15. 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?

  16. How to Solve • Step1: Set up the equation 64x+45y=$63 5x=3yStep2: Choose a method. In this case, substituting would be the best choice. 5x=3y 3 3 5x =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= $0.45Step4: Write down the answer in complete sentences. Each orange would cost a total of $0.45

  17. 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?

  18. How to solve Step 1: Set up the equations. x+y=500 4x+3y=1600 Step 2: Multiply one variable in an equation to equal the other equation. 4(x+y)=500 4x+3y= 1600 Step 3: Distribute and Solve. 4x+4y=2000 4x+4y=2000 4x+3y=1600  - 4x+3y=1600 y=400 Step 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.

  19. 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. • Write two equations using the information above. • What is the total Necklace Sally and John both sold? • How much gift bags were sold from each? • Graph both equations and decide which company makes more money.

  20. How to Solve A) Step one, plug what is given into the formula C=ax+by Sally: $7520= $375X + 2B John: $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

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

  22. 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

  23. 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. • Using the points (2,30) and (3,35) find the slop of Joe’s equation. • Using the points (2,24) and (4,28) find the slope of sally’s equation. • Find the y-intercept of the 2 equations, if Joe made a total of $45 and Sally made a total of $30.

  24. 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 +b 15 = 3x -25 -25 3 3 20 = b  solve for b. 5=x

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