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1. Describe each transformation of f to g: f(x) = |x| and g(x) = -3|x + 2| - 1. 2. Write a function g if f(x) = x2 has a vertical shrink of 1/3
followed by a translation up 2.
2. The data shows the humerus lengths ( in centimeters) and heights (in centimeters( of several females.
Use the graphing calculator to find a line of best fit for the data.
Estimate the height of a female whose humerus is 40 centimeters long. Estimate the humerus length of a female with a height of 130 cm.
Algebra II 1
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Systems of Equations with Two Variables
Algebra II
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two or more linear equations.
Looks like
A solution is an ordered pair that makes all equations true.
Algebra II 3
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3x – 2y = 2 x + 2y = 6
a) (0, -1)
b) (2,2)
no
yes
Algebra II 4
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GraphingSubstitutionElimination
5Algebra II
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To find the solution of a system of two linear equations: (steps)
1. Graph each equation2. Identify the intersection3. This is the solution to the system
because it is the point that satisfies both equations.
**Remember that a graph is just a picture of the solutions.
Algebra II 6
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coincident lines(same line)
infinitely many solutions
parallel linesno solutions
intersecting linesone solution
Graph Number of Solutions
Two lines intersect at one point.
Parallel lines
Lines coincide
Algebra II 7
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First, graph 2x – 2y = -8.
Second, graph 2x + 2y = 4.
The lines intersect at (-1, 3)
The solution is (-1, 3)
Solve the system of equations by graphing.
Algebra II 8
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First, graph -x + 3y = 6.
(0, -1)(3, 0)Second, graph 3x – 9y = 9.
(0, 2)
(-3, 1)
The lines are parallel.
No solution
(3, 3)
Solve the system of equations by graphing.
(-3, -2)
Algebra II 9
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First, graph 2x – y = 6.
Second, graph x + 3y = 10.
The lines intersect at (4, 2)
The solution is (4, 2)
Solve the system of equations by graphing.
Algebra II 10
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First, graph x = 3y – 1. (-1, 0) (2, 1)
Second, graph 2x – 6y = -2.
The lines are identical.
Infinitely many solutions
Solve the system of equations by graphing.
(-4, -1)
Algebra II 11
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Steps for Substitution:
1. Solve one of the equations for one variable (try to solve for the variable with a coefficient of one)
2. Substitute the expression into the other equation and solve the new equation.
3. Substitute the value from step 2 into one of your original equations to complete the ordered pair
Algebra II 12
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Algebra II 13
(2,0)
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Algebra II 14
(1/4, -5/4)
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Algebra II 15
Infinitely Many Solutions
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Algebra II 16
(-3,-6)
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Steps for elimination:
1. Make one of the variables have opposite coefficients (multiply by a constant if necessary)
2. Add the equations together and solve for the remaining variable
3. Substitute the value from step 3 into one of the original equations to complete the ordered pair
Algebra II 17
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Solve the following system by
elimination
6x – 3y = –34x + 5y = –9
30x – 15y = -1512x + 15y = -2742x + 0 = -42
42x = -4242 42
x = -11
2
5(6x – 3y = –3)3(4x + 5y = –9)
Algebra II 18
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Use x = -1 to find y
2nd equation: 4x + 5y = -94(-1) + 5y = -9
-4 + 5y = -9+4 +4
5y = -55 5y = -1
(-1, -1)
3
Algebra II 19
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Solve the following system by
elimination
3x – y = 46x – 2y = 4
-6x + 2y = -86x – 2y = 40 + 0 = -4
0 = -4False!
No Solution
1
2
-2(3x – y = 4)(6x – 2y = 4)
≠
Algebra II 20
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Solve the following system by
elimination 3x + 5y = -6
2x – 2y = -8
6x + 10y = -12-6x + 6y = 240 + 16y = 12
16y = 1216 16y = 3/4
1
2
2(3x + 5y = -6)-3(2x – 2y = -8)
Algebra II 21
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Use y = 3/4 to find x
1st equation: 3x + 5y = -63x+ 5(3/4) = -63x + 15/4 = -6
-15/4 -15/4 3x = -39/4
3 3y = -13/4
(-13/4, 3/4)
3
Algebra II 22
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Solve the following system by elimination
-2x + y = -58x – 4y = 20
-8x + 4y = -208x – 4y = 20
0 + 0 = 00 = 0True!
Infinitely Many Solutions
1
2
4(-2x + y = -5)(8x – 4y = 20)
=
Algebra II 23
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24Algebra II
1. 4x – 3y = 10 2x + 2y = 7
2. Y = 3x – 5 2x + 3y = 8
3. X – 3y = 10 4x + 3y = 21
4. 3x + 2y = 8 2y + 4x = -2
5. 2x + 7y = 10 x + 4y = 9
6. x – 3y = -6 x = 2y
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1. 4x – 3y = 10 8x – 6y = 5
2. 3x + 3y = 10 2x – 2y = 15
M = 4/3, b= -10/3
M = 4/3 b = -5/6
No solution
M = -1, b = 10/3
M = 1, b = -15/2
One solution
Algebra II 25
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3. y = 2x + 8 2x – y = -8
4. 1/2x + 3y = 6 1/3x – 5y = -3
M = 2, b= 8M = 2, b = 8Infinitely many
M = -1/6, b = 2
M = 1/15, b = 3/5 One solution
Algebra II 26
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1. Your family is planning a 7 day trip to Florida. You estimate that it will cost $275 per day in Tampa and $400 per day in Orlando. Your total budget for the 7 day is $2300. How many days should you spend in each location? X = # of days in TampaY = # of days in Orlando
X + y = 7275x + 400 y = 2300
27Algebra II
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2. You plan to work 200 hours this summer mowing lawns or babysitting. You need to make a total of $1300. Babysitting pays $6 per hour and lawn mowing pays $8 per hour. How many hours should you work at each job? X = # of hours babysittingY = # of hours of mowing
X + y = 2006x + 8y = 1300
28Algebra II
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3. You make small wreaths and large wreaths to sell at a craft fair. Small wreaths sell for $8 and large wreaths sell for $12. You think you can sell 40 wreaths all together and want to make $400. How many of each type of wreath should you bring to the fair?X = # small wreathsY = # large wreaths
X + y = 408x + 12y = 400
29Algebra II
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4. You are buying lotions or soaps for 12 of your friends. You spent $100. Soaps cost $5 a piece and lotions are $8. How many of each did you buy?
x = # of soapsy = # of lotions
x + y = 12 5x + 8y = 100
Algebra II 30
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5. Becky has 52 coins in nickels and dimes. She has a total of $4.65. How many of each coin does she have?
x = # of nickelsy = # of dimes
x + y = 52 .05x + .10y = 4.65
Algebra II 31
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6. There were twice as many students as adults at the ball game. There were 2500 people at the game. How many students and how many parents were at the game?
x = # of studentsy = # of parents
x = 2y x + y = 2500
Algebra II 32
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1. Using substitution, solve the system:
3x + 4y = -4 x + 2y = 2
2. Using elimination, solve the system:
-3x + y = 11 5x – 2y = -16
{
{ (-6, -7)
(-8, 5)
Algebra II 33