Download Mastery Learning Algebra 1 Systems of Equations, Direct Variation

Mastery Learning Algebra 1
Systems of Equations, Direct Variation, Square Root
Systems of Equations
Solve by matrix:
Write both equations with X first then Y (on the same
side)
You Try
1. Given
2x  3y  12
6x  2y  42
What is x+y?
Ax+By=C
CALCULATOR: [2nd] [MATRIX]->2x3->Enter
coefficients of all six[2nd][QUIT]->[2nd][MATRIX]>MATH->B:rref(->]->[2nd][MATRIX]A[ENTER]
Types of Solutions:
1 0 # x 
  (# x ,# y )
0 1 # y 
1 0 # 
NO SOLUTION: 

0 01 
1 0 # x 

MANY SOLUTIONS: 

0 0 0 
ONE SOLUTION: 
If a variable
 is missing then put a ‘0’
If the number in front of a variable is missing then
put a ‘1;
Ex: x = 3y -
2 & 4x = 16
1x – 3y = -2
4x + 0y = 16
1  3  2 
and then do RREF[A]
16 
4 0
1 0 4 
and you get 
 so x = 4 and y = 2
0 1 2 
So [A] is 
Solve by Graphing:
Solve for y, graph find intersection.
Direct Variation y=kx
 K is the constant of variation, plug in values
for x and y to solve for k. Then use second
value for x or y to solve for missing variable.
 EX: If y varies directly as x, and y=15 when
x=-60, then what is the value of y when
x=256?
 Use y=kx 15=-60k



restaurant received 550 hamburgers and 425
hotdogs for $630.
a. How much did each hamburger cost?
b. How much will 25 hamburgers and 50 hotdogs be?
3. A local pet store has triple the amount of fish as
birds and has a total of 250 fish and birds. Write a
system of equations that represents the number of
fish and birds using the variables F and B.
4. Given the system of equations
4x  3y  60
x  y 10
what
is the value of x?
5. Given:
A. 11
2x + y = 15
5x - 6y = -22 What is the value of x - y?
B. 2
C. 3
D. -3
6. Given: w = 1 - v
2v + w = 4 Find the value of w.
A. 3
B. 2
C. 1
D. -2
7. A limousine company charges a flat-fee of $80 plus
$.05 per mile. A shuttle van company charges a
flat-fee of $60 plus $.50 per mile. Approximately
what mileage will yield the same fare for both?
A. 24 miles B. 34 miles C. 44 miles D. 54 miles
8. If y varies directly as x, and y=20 when x=-2, then
what is the value of y when x=15?
9. If y varies directly as x, and y = 10 when x = 4, then
what is the value of x when y = 25.
10. The number of gallons of gasoline varies directly
with the cost per gallon. It costs $75 for 20 gallons
1
of gasoline. How much does it cost for 13 gallons?
k=  So…substitute the value of k into y=kx
4
with given value of x

2. A restaurant received 270 hamburger patties and
 350 hotdogs on Monday for $450. On Friday the
1
4
y=256(  )
y=-64
11. A factory can produce 123 radios in 2 weeks.
Assuming direct variation how many radios can be
produced in 5 weeks?
Mastery Learning Algebra 1
Systems of Equations, Direct Variation, Square Root
Square Root
Simplifying Radical Expressions: We don’t like
decimals all the time.
 Find the prime factorization of the radicand
 SQUARE ROOTS are like a house with party
on the inside and only couples (pairs) can
leave together.
o All pairs of like numbers or variables
are multiplied in front of the square
root.
o All the single numbers or variables
remain inside the square root.
o If no singles remain, then don’t write
square root. The radicand is called a
perfect square.




EX: 136 Prime Factorization of
136= 2  2  2 17 . There is one pair of 2’s
ANSWER: 2 2 17  2 34
12. The length of a spring varies directly as the
weight attached. If a spring stretches 1.6 inches
when a 24 pound weight is attached, how far will
the spring stretch when a 15 pound weight is
attached?
A. 3.9 inches
B. 2.56 inches
C. 2.25 inches D. 1 inch
13. Police officers can use the formula s  30 fd to
determine the speed s that a car was traveling in
miles per hour by measuring the distance d in feet of
its skid marks. In this formula, f is the coefficient of
friction for the type and condition of the road.
a. Write a simplified expression for the speed if f =
0.6 for a wet asphalt roaD.
b. What is a simplified expression for the speed if f =
0.8 for a dry asphalt road?
c. An officer measures skid marks that are 110 feet
long. Determine the speed of the car for both wet
road conditions and for dry road conditions.
14. Find the area of the rectangle if l=( 3 ) and
w=( 12  5 ).
A.
EX:
5
C. 6 3  3 5
2
72 x y

72  2  2  2  3  3
x5  x  x  x  x  x
y2  y  y
Calculator method for multiple choice: Type in the
original problem and write down the decimal. Then
check all the answer choices to see which one also
gives the same decimal answer as the problem.
The Distance formula
d  ( x 2  x1 ) 2  ( y 2  y1 ) 2
(( x2  x1 ) 2  ( y2  y1 ) 2
B. 6 +
D.
15
51
15. The side of a square is connected between
(1,-2) and (-3, 1). Find the square’s perimeter.
(Use the distance formula to find one side)
A.
A. 20 units
B. 25 units
ANSWER  6 x 2 y 2 x
Calculator: d 
36  15
C. 4 7 units
D. 7 units.
16. Voltage V is given by V  PR where P is the
power in Watts and R is the resistance in ohms.
How many more volts are needed to light a 75-watt
bulb than a 50-watt bulb if the resistance for both is
100 ohms? (Hint: Which letters of the equation can
substitute for numbers?)
17. Simplify:
A.
2 54 x 4 y 7
6 x 2 y 3 6 y B. 2 x 2 y 3 6 y
C. 2 x
2
6y7
D. 2