Download MPM 2D1 – MATHEMATICS REVIEW – PART 2

MPM2D – MATHEMATICS REVIEW – PART 2
1)
Evaluating Expressions and Formulas
- these are evaluated by substituting the given numbers in place of the variables.
They follow the order of operations.
Ex.
Find the value of 2x + y if x = 2 and y = –3
2x + y = 2(2) + (–3)
=4–3
=1
Now try 2x 2 – y if x = –2 and y = 3.
2x 2 – y = 2(–2)2 – 3
= 2(4) – 3
=8–3
=5
2)
Algebraic Expressions
- contain both numbers and letters. The numbers are called coefficients and
the letters are called variables.
3)
Terms
- a combination of numbers and variables multiplied together.
Ex.
–3xy 2
- in an algebraic expression, terms are separated by a positive or negative sign.
Ex.
5x 2 + 3x – 12 has three terms
- an algebraic expression with one or more terms is called a polynomial.
- monomial: one term
Ex. 8x
- binomial: two terms Ex. 5x – 9
- trinomial: three terms Ex. 6x + 4y + 10
4)
Adding and Subtracting Algebraic Expressions (collecting like terms)
- algebraic expressions can be added or subtracted only if we have like terms,
that is, terms that have the same variable and the same exponent.
Ex.
Like terms: 5x 2, 3x 2, –24x 2
Unlike terms: 5x 2, 5x, 5y 2
Try: 2x 2 + 3 – 4x 2 + 8
= 2x 2 – 4x 2 + 3 + 8
= –2x 2 + 11
5)
Simplifying Algebraic Expressions
- first you must remove the brackets
- if the expressions are being added, then simply remove the brackets
- if the expressions are being subtracted, you need to change all of the signs
in the bracket following the negative sign before you remove the brackets
Ex.
(4x – 5y) + (6x + 3)
= 4x – 5y + 6x + 3
= 4x + 6x – 5y + 3
Ex.
= 10x – 5y + 3
6)
(4x + 9y) – (5x – 7y)
= 4x + 9y – 5x + 7y
= 4x – 5x + 9y + 7y
= –x + 16y
Distributive Property
- algebraic expressions can be multiplied by a constant
- in order to remove the brackets, you must multiply the constant by each
term inside the bracket
Ex.
2(3x – 5y + 2)
= 2(3x) + 2(–5y) + 2(2)
= 6x –10y + 4
Ex.
–3(4x – 3y) + 2(2x + 5)
= –3(4x) + (–3)(–3y) + 2(2x) + 2(5)
= –12x + 9y + 4x +10
= –12x + 4x + 9y +10
= –8x + 9y +10
7)
Solving Equations
a) An equation is a mathematical statement that states that two quantities are
equal.
b) A solution or root to an equation is a number which makes the left side
equal to the right side.
c) In order to solve an equation, you must isolate the variable, that is, have all
of the variables on one side and all of the numbers on the other side.
Variables and numbers can be moved from side to side by performing the
opposite operation to both sides of the equation.
d) All brackets must be eliminated by using the distributive property.
e) All fractions must be eliminated by multiplying each term of the equation
by the lowest common denominator.
Ex.1:
13x + 9 = 11x + 5
13x – 11x = 5 – 9
2x = –4
Ex. 2: y + 6(y – 3) = 2(3y – 2)
y + 6y – 18= 6y – 4
y + 6y – 6y = 18 – 4
x = –2
Ex. 3:
1 x 7
 
2 3 6
6(1) 6(x) 6(7)


2
3
6
6 6x 42


2 3
6
3  2x  7
2x  7  3
2x  4
x2
y = 14
Ex. 4:
7
1
x2
6
3
6( 7 )
6(1)
x  6(2) 
6
3
42
6
x  12 
6
3
7x  12  2
7x  2  12
7x  14
x2