Pre-Calculus 11 By: Kate Reid Systems of Equations Chapter 8 Linear and Quadratic Inequalities Chapter 9 Solving Systems of Equations Graphically 8.1 Solving Systems of Equations Algebraically 8.2 Types Of Systems System of Linear-Quadratic Equations System of Quadratic-Quadratic Equations a linear equation and a quadratic equation involving the same variables a graph of the system involves a line and a parabola y=x-2

y=x -4x-2 2 two quadratic equations involving the same variables the graph includes two parabolas y=x -4

y=x 2 2 Solving Possible Numbers of Solutions One Zero Two One per intercept no intercepts one intercept two intercepts Answer x and y values of intercepts System of Equations a collection of two or more equations with the same variables Method 1 Substitution Method 2 Elimination Step 1 Make Into One Formula with One Variable Step 2 Solve remove a variable by replacing a variable with its equivalent 5x-10=y

x +x-2y=0 2 x +x-2(5x-10)=0

x -9x+20=0 5x-y=10

x +x-2y=0 2 2 remove variable by subtracting the equations -2(5x-y)=-2(10)

=-10x+2y=-20 -10x+2y=-20

x +x-2y=0 2 2 Step 3 Find the y Value(s) Step 4 Check - x -9x=-20 2 Method 1 Factoring Method 2 Quadratic Formula find x value(s) x -9x=-20 2 x -9x+20=0

x -4x-5x+20=0

x(x-4)-5(x-4)=0

(x-4)(x-5)=0

x=4,5 2 2 x -9x+20=0 2 ax +bx+c=0 -(-9)+[(-9) -4(1)(20)] 1/2 2 _ ---------------------------- 2(1) 9+1

2 = ----- =4,5 plug x values into formula and solve for y x=4

5x-10=y

5(4)-10=y

20-10=y

y=10 x=5

5x-10=y

5(5)-10=y

25-10=y

y=15 plug both x and y values into other formula (4,10)

x +x-2y=0

(4) +(4)-2(10)=0

16+4-20=0

0=0 correct (5,15)

x +x-2y=0

(5) +(5)-2(15)=0

25+5-30=0

0=0 2 2 2 2 correct Linear Inequalities in Two Variables 9.1 Quadratic Inequalities in One Variable 9.2 Quadratic Inequalities in Two Variables 9.3 Inequality a statement comparing expressions that may not be equal Symbols = > < < > _ _ Solution Region all the points in this region satisfy the inequality Boundary the line that separates the regions dashed line b/c the numbers on the line aren't answers(<) Test Point a point that is used to det. the region that satisfies the inequality plug into inequality to check Solution Region all the points in this region satisfy the inequality Boundary the line that separates the regions dashed line b/c the numbers on the line aren't answers(>) Test Point a point that is used to det. the region that satisfies the inequality plug into inequality to check plug into inequality to check a point that is used to det. the region that satisfies the inequality Test Point Boundary Solid line b/c the numbers on the line are included in the answer(<) Solution Region all the points in this region satisfy the inequality _ Test Point Plug into inequality to check a point that is used to det. the region that satisfies the inequality Solution Region all the points in this region satisfy the inequality b/c the numbers on the line are included in the answer(<) Boundary Solid Line Boundary Solution Region all the points that satisfy the inequality Solid line b/c the numbers on the line satisfy the inequality(>) Test Point a point that is used to det. the region that satisfies the inequality Boundary Solid Line _ b/c the numbers on the line satisfy the inequality(<) a point that is used to det. the region that satisfies the inequality Test Point Solution Region _ all the points that satisfy the inequality a point that is used to det. the region that satisfies the inequality Test Point Solution Region all points that satisfy the inequality(>) Boundary Dashed Line b/c the numbers on the line don't satisfy the inequality b/c there are 2 variables the answer will be a point not just an x value Test Point Solution Region Boundary Dashed Line _ all points in the region satisfy the inequality(<) _ the line that separates the regions the line that separates the regions the line that separates the regions the line that separates the regions the line that separates the regions the line that separates the regions b/c the numbers on the line don't satisfy the inequality a point that is used to det. the region that satisfies the inequality Domain:(-4<x<2) Range:(y>-3) _ _ _ Domain:(- ,-6] u[6, ) 8 Range:[6, ) 8 Domain:(-1<x<5) Range:(y>-3) Domain:(- ,-4)u(0, ) 8 Range:(y<2) b/c there is 1 variable the answer will only be an x value Solve with Algebra Domain/Solution:(-4<x<0) Range:(y<2) uncoloured circles don't include points Domain/Solution:(- ,-1)(5, ) 8 8 8 8 Range:(y>-3) Domain/Solution:(-4<x<2) Range:(y>-3) uncloured circles don't include points coloured circles include points _ _ Domain/Solution:(- ,-6]u[6, ) 8 8 Range:(y>-4) A

x<-1 B

-1<x<3 C

x>3 x -2x-3<0 _ plug in an x value from each section of the line 2 Section A

(-2) -2(-2)-3<0

4+4-3<0

5<0

Incorrect doesn't satisfy the inequality Section B

(1) -2(1)-3<0

1-2-3<0

-6<0 Correct, satisfies the inequality Section C

(4) -2(4)-3<0

16-8-3<0

5<0 Incorrect doesn't satisfy the inequality _ _ _ _ _ _ 2 2 _ _ _ 2 _ _ _ _ Answer:(-1<x<3) Final Installment

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# Pre-Calculus 11

Chapters 8 and 9