Download REVIEW

MPM2D Review
GRADE 9 REVIEW
A.
NAME:
OPERATIONS WITH RATIONAL NUMBERS
Addition & Subtraction



a)
In order to add and subtract when working with fractions, must have a COMMON
DENOMINATOR.
Convert MIXED FRACTIONS to IMPROPER FRACTIONS.
Reduce to lowest terms.
2 1

7 2
b) 
2
5
2
3
6
2
7
c)  1  3
5
10
d)
12 2

5 3
Multiplication & Division




When multiplying fractions, you do not need a common denominator.
Multiply the numbers across the top and multiply the numbers across the bottom.
When dividing fractions: multiply and use the reciprocal of the second fraction (flip).
Reduce to lowest terms.
 7  17 

 2  3 
a)   
4 2
b) 
7 5
2 1
c) 3 
3 4
 2   3 
 

3  4 

d)
1
2
2
1
MPM2D Review
B.



ALGEBRAIC EXPRESSIONS (POLYNOMIALS)
ALGEBRAIC EXPRESSIONS contain both numbers and variables (letters)
Algebraic expressions with one or more terms are called POLYNOMIALS.
Algebraic expressions can only be simplified if they contain LIKE TERMS (this means that all of the
variables and the exponents on each variable are exactly the same).
Example:
3x2 and -5x2  Like terms
2xy and 7yz  Unlike terms
Example 1:
Simplify each of the following:
a. 2 x  3 y  5 x  7 y
2
2
b.
4x2  6 y  x2  7 y
Example 2:
a. Find the value of 2 x 2  y
Find the value of 3 x  2 y
b.
if x  
if x  2 and y  3

1
5
and y 
2
3
When multiplying algebraic expressions use the DISTRIBUTIVE PROPERTY.
Example 3:
Expand & Simplify
a. 24x  3  4x  5
b.

 
3 x2  7  2 6x2  x  6

2
MPM2D Review
C.

LINEAR RELATIONS
A linear equation is an equation with a degree of 1 (there are no exponents)
i.e. 2x + 4y – 8 = 0
STANDARD FORM
 A linear equation in STANDARD FORM looks like Ax  By  C  0 , where A, B, and C are integers.
 To put into standard form:
 ALL terms are on the left side of the equation and zero is on the right side
 ‘a’ in the equation is always a positive value
 Eliminate ALL fractions
Example 1:
a)
Express each equation in standard form
y  3 x  10
b)
x5
 2y
3
SLOPE Y-INTERCEPT FORM
 A linear equation in SLOPE Y-INTERCEPT FORM looks like y  mx  b , where “m” represents the
slope of the line and “b” represents the y-intercept
 To put into slope y-intercpet form:
 Isolate the y on the left side of the equation
 ‘y’ in the equation is always positive
Example 1:
Put the following into slope y-intercept form
a. 3 x  2 y  6
b.
4 x  y  8  10
3
MPM2D Review
D.
FINDING SLOPE
SLOPE – The rate of change of a linear relationship

There is more than one way to represent the slope:
m
Example:
rise
run
m
OR
y 2  y1
x 2  x1
Find the slope of the line passing through the following points:
b.
(3, –6) and (1, 5)
a.
(4, 6) and (7, 9)
E.
GRAPHING LINEAR RELATIONS
c.
(–1, –4) and (–3, 8)
There are three different ways that you can graph a linear relation.
1. Table of values
2. x & y intercepts
3. Slope y-intercept form
1.



TABLE OF VALUES
Choose any 5 values for x and sub into the equation
Solve for y
Graph the ordered pairs and connect them with a solid straight line that extends past each end point
Example:
x
Using a table of values, graph the equation y  2 x  4
y
-2
-1
0
1
2
4
MPM2D Review
2.



X and Y INTERCEPTS
This method gives you two ordered pairs to graph.
The x-intercept is the point where the line crosses the x-axis and its coordinates are (x, 0). Let y = 0 to
solve for x.
The y-intercept is the point where the line crosses the y-axis and its coordinates are (0, y). Let x = 0 to
solve for y.
Example 1:
a)
Determine the x and y intercepts for each equation.
2x  6 y  5
Example 2:
b)
4 x  3 y  12  0
Using the x and y intercepts, graph the equation
2 x  3 y  12
5
MPM2D Review
3.
SLOPE Y-INTERCEPT FORM
y  mx  b
To graph using this form:
1. Put equation in the form y  mx  b
2. Plot the y-intercept on the y-axis
3. From the y-intercept use the slope to determine the location of the second point
(rise over run)
Example:
F.


Determine the slope and y-intercept and graph:
y  2x  4
FINDING THE EQUATION OF A LINE
When given a line in the form y = mx + b, we can easily graph the linear relation, and easily determine
the slope and the y – intercept.
When the slope and y-intercept are not given, we need to find the slope and y-intercept in order to
determine the equation of the line.
To find the equation of the line:
1) Find the slope using the formula
2) Find the y-intercept by substituting the coordinates from a point on the line and slope into y = mx +
b to solve for b
3) Write the equation in y = mx + b form.
Example 1:
Find the equation of the line when A(-6, -2) and B(4, 3)
6
MPM2D Review
Example 2:
G.
Find the equation of a line if the x-intercept is 5 and the y-intercept is –2.
PARALLEL AND PERPENDICULAR SLOPES



Parallel lines have the exact same slope
Perpendicular lines intersect each other at 90◦
Perpendicular lines have slopes that are negative reciprocals of each other.

Horizontal lines have no x-intercept and are parallel to the x-axis.
 The equation of horizontal line is y = a (0, a)
 The slope is always zero

Vertical lines have no y-intercept and are parallel to the y-axis
 The equation of a vertical line is x = b (b, 0)
 The slope is always undefined
Example 1:
Write the equation of the line that is parallel to y = -4 and passes through the point (5, 1).
Example 2:
Write the equation of the line that is parallel to the line y = 3x + 17 and passes through the
point (7, -6).
7
MPM2D Review
Write the equation of the line that is perpendicular to y 
Example 3:
3
x  6 and passes through the
5
point (9, 6).
H.
EXPONENT LAWS
Exponent Laws
Examples
In general:
a m  a  a  a  ...  a m times
34  3  3  3  3  81
Multiplication:
a m  a n  a m n
x 3  x 2  x 32  x 5
Division:
am
 a mn
n
a
x7
 x 7 3  x 4
3
x
Power Law:
a 
 a mn
x 
Power of a Product:
ab 
 a b
Power of a Quotient:
am
a
   m
b
b
m n
m
m
5 2
xy3  x 3  y 3  x 3 y 3
2 x 4  2 4  x 4  16 x 4
m
m
1. 33
x 
2 5
2.
3
1 3
5.     
4 4
2
4
 
6. x xy
3 2
 x10
2
x2 x2
 x
   2 
2
4
2
3.  10 
4.  10 
2
 x4 y3 
7.  6 
 xy 
2
8.
2
xy 
2 3
8