7.1Solvingsys2015.notebook. November 05, Warm up. Partial fraction decompostion

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1 Warm up Partial fraction decompostion 1

2 Please add due dates to the calendar Nov Dec 2

3 7.1 Solving Systems of Equations by Substitution and Graphing Vocabulary System: Problems that involve two or more equations in two or more variables. Solution: An ordered pair that satisfies each equation in the system. The solution of a system of equations corresponds to the Points of intersection of the graphs of the equations in the system. 3

4 Methods to Solve Systems Substitution Method Graphically Elimination Method (next section) 4

5 Substitution Method 1. Solve one of the equations for one variable in terms of the other. 2. Substitute the expression into the other equation to obtain an equation all in terms of one variable. 3. Solve the equation. 4. Back substitute to find the other variable. 5 Check solution...it must satisfy both original equations. 5

6 Solve the system using substitution: 2x + y = 2 x 2y = 9 Step 1: y = 2x +2 Step 2: x 2( 2x+2) = 9 Step 3: x 2( 2x+2) = 9 x + 4x 4 = 9 5x 4 = 9 5x = 5 x= 1 Check: 2( 1) + (4) = 2 ( 1) 2(4) = 9 Step 4: 2x + y = 2 2( 1) + y = y = 2 y = 4 6

7 Solve the system using substitution: y = x 3 3x y = 2x + 4 x 3 3x = 2x + 4 x 3 3x 2 +2x = 0 x(x 2 3x+ 2) = 0 x ( x 2)(x 1)=0 x=0, x=2, x=1 y = 2x + 4 y = 2(0)+4 y = 4 (0,4) y = 2x + 4 y = 2(2)+4 y = y = 0 (2,0) Set equal to each other since both are equal to y. Factor Substitute back into one of the equations to find all the solutions. y = 2x + 4 y = 2(1) + 4 y = y = 2 (1,2) If you graphed it, you would see it is intersecting in 3 points. 7

8 Solve the system using substitution: x + y = 4 x 2 + y = 3 y = x + 4 x 2 + (x + 4 )= 3 x 2 + x + 1 = 0 Imaginary, so no solution If you graph it, you will see they will never intersect. 8

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10 Solve the system of equations Graph each equation and find the point of intersection 10

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12 Graphically The solution of a system of equations corresponds to the point of intersection of the graphs of the equations in the system. 2x + y = 6 x + y = 0 Solution: (2,2) 12

13 Solve graphically: 3x 2y = 0 x 2 + y 2 = 4 Solution: (1.109, 1.664) ( 1.109, 1.664) 13

14 Points of Intersection and Applications Business Total cost C typically has two components: 1. initial cost 2. cost per unit Break even point: When revenue = costs (you are not making money, you are not losing money) P = R C When R = C then P (profit = zero) Does that make sense? 14

15 The total cost C of producing x units of a product typically has two components: The initial cost and the cost per unit In break even analysis, the break even point corresponds to the point of intersection of the cost and revenue curves. Break even analysis can also be approached from the point of view of profit. In this case, consider the profit function, which is P= R C. The break even point occurs when profit equals zero or when R=C. 15

16 Applications A small business invests $10,000 in equipment to produce a new soft drink. Each bottle costs $0.65 to produce and is sold for $1.20. How many bottles must be sold before the business breaks even? 16

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18 A company has a fixed monthly manufacturing cost of $12,000, and it costs $0.95 to produce each unit. The company sells each unit for $1.25. How many units must be sold before the company breaks even? C = x R = 1.25 x R= C x = 1.25x =.3x x = They need to sell 40,000 buttons. 18

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