Download 4 - Connell Math

MPM1D
Date: ________________
4.1 Solve Simple Equations
To maintain equilibrium in an equation you must
perform the same operation on both sides of the
equation.
To ISOLATE a variable you need to perform the
opposite operation. Starting backwards from
BEDMAS... SAMDEB
Undo additions & subtractions first
then multiplications and divisions.
What is the opposite operation?
addition------->
subtraction----->
multiplication---->
division-------->
exponent 2------>
square root----->
Ex: Solve
a) m + 3 = 4
Communication:
Keep equal signs aligned, only
one per line
MPM1D
Date: ________________
b) x - 2 = 6
c) 4x = 20
d)
x
 12
3
2
e) x  36
MPM1D
Date: ________________
Always remove the constants before removing the coefficients!
f) 2x - 11 = 27
Check your solution to example 2 f)
Checking a solution
Substitute the root into the right and left
side of the equation. Both sides must be
equal.
Communication:
You must separate the Left Side (LS)from
the Right Side (RS) of the equation.
g) 5 
x
3
2
MPM1D
h) 3 x 
Date: ________________
2
 4
5
i) 5m 
4
2
3
Ex 3: At a computer store, packages of DVDs sell for $15 each.
a) Write an equation to model the number of packages of DVDs
bought.
b) Given that one customer buys $120 worth of DVDs use your
equation to solve for the number of packages of DVDs.
Let
represent the number of packages of DVDs
MPM1D
Date: ________________
4.2 Solve Multi-Step Equations
Recall: To solve equations "undo" additions & subtractions first,
then multiplications & divisions.
To solve equations with variables on both sides, use inverse
operations to group the variable terms on one side of the equation.
Ex. 1 Solve
a) 3x - 7 = 8x + 8
b) 2x + 8x - 4 = 6x + 10 - 8
c) 5(x + 4) = 3x + 14
MPM1D
d) 7 - ( 4m + 3 ) = -3 (m + 2) - (2m + 3)
Ex. 2: Check your solution to Ex. 1 d)
Date: ________________
MPM1D
Date: ________________
Ex. 3: Two or more angles are supplementary if their sum is 180
degrees. An angle is 4 times the value of its supplement. Set up
and solve an equation to find the measures of the two angles.
Ex. 4: The square and equilateral triangle shown have the same
perimeters. What are the dimensions of each figure?
MPM1D
Date: ________________
4.3 Solve Equations Involving Fractions
Eliminate fractions by multiplying every term by
the common denominator
Ex. 1 Solve
a)
x
2
6
b)
1
a  6  7
2
c)
x 1
7
3
Communication:
Keep equal signs aligned,
only one per line
MPM1D
d)
m m
 2
2 3
e)
3  2m m

4
3
f)
x 3x  1
x

3
2
4
6
Date: ________________
MPM1D
g)
Date: ________________
2
1
(x  5)  (x  2)  2
3
4
Ex. 2: A triangular backyard has dimensions as shown.
a) Write an algebraic expression for the perimeter.
b) Determine the dimensions of the yard if the perimeter is 60 m.
MPM1D
Date: ________________
4.4 Modelling with Formulas
Formulas can be rearranged to isolate different variables.
Isolate the variable that is
requested! Use the same process
used to solve equations.
Ex. 1 Solve for the variable indicated.
a) y  mx  b solve for b
b) y  mx  b
solve for m
2
c) V   r h
solve for r
solve for h
2
d) A   r
MPM1D
e) 4 x  3y  15 solve for y
Date: ________________
f) 2 x  8y  10
Ex. 2 The formula for the area of a trapezoid is A 
solve for y
h
(a  b)
2
Solve for b then find the length of side b if the area is 84 cm2 , the
height is 8 cm, and the length of side a is 12 cm.
MPM1D
Date: ________________
4.5A Modelling with Algebra
1. State the correct operation
twice
___________
half
___________
difference
___________
warmer
___________
product
___________
quotient
___________
fewer
___________
is
___________
times
___________
older
___________
reduced
___________
younger
___________
are
___________
more
___________
triple
___________
quarter
___________
taken from
___________
gives us
___________
2. If n represents a number, write an algebraic expression using
numbers and symbols for each of the following statements.
a) Three times a number
_______________________
b) A number increased by one
_______________________
c) A number decreased by five
_______________________
d) Four more than twice a number
_______________________
e) Half a number
_______________________
f) Double a number reduced by six
_______________________
MPM1D
Date: ________________
Number Problems
1) A number divided by 2, increased by 6 is 11. Find the number.
2) One number is 2 more than 3 times another number. If the sum of
the numbers is 14, what are the numbers?
MPM1D
Date: ________________
3) Double the square of a number is increased by 2 resulting in 52.
What is the number?
4) Find two consecutive numbers with a sum of 149.
Consecutive Numbers:
x, x + 1, x + 2, ...
5) The sum of two numbers is 34. The difference between the two
numbers is 4. Find the two numbers.
MPM1D
Date: ________________
4.5 B Modelling with Algebra
Measurement Porblems
Ex. 1 The sides of a triangle are 3 consecutive whole numbers. The
perimeter of the triangle is 48 cm. How long is each side?
Let
represent the smallest number.
Ex. 2 The length of a rectangle is 3 m greater than the width. The
perimeter is 26 m. What are the dimensions of the rectangle?
MPM1D
Date: ________________
Money Problems
Ex. 3 Heather earned $3 more than double the amount Oliver earned.
The difference of their earnings was $15. How much did each person
earn?
Ex. 4 A parking meter contains $27.05 in quarters and dimes. There are
146 coins. How many quarters are there?
MPM1D
Date: ________________
Other Word Problems
Ex. 5 Farren's mother is 4 years older than twice Farren's age. The
difference of their ages is 22 years. Find their ages.
Ex. 6 Two cars left the same highway restaurant at the same time but
drove in opposite directions. One travelled at 65 km/h, the other at 55
km/h. After how long were they 600 km apart?