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Algebra 1
First Semester Exam Review
Chapter 1—Connections to Algebra
1.1 Variables in Algebra
Variable—a letter to represent one or more numbers
Ex. x, y, n
Values—the numbers of a variable
Ex. 4, 78, etc.
Expression—a mathematical statement with numbers, variables, and operations
Equation—expression joined by an =
Ex. 14x = 35
Ex. 3x – 2y
1.2 Exponents and Powers
Exponent—the number of times a base is used as a factor
Base—the number to be multiplied by itself in a power
Power—an expression using a base and an exponent
Ex. 53, base = 5,
exponent = 3
power = 53 = 5  5  5 , not 5 3
1.3 Order of Operations
Order of Operations—to evaluate expressions with more than one operation
P—Parenthesis (symbols of inclusion: ( ), [ ], { })
E—Exponents (powers)
M—Multiplication (left to right)
D—Division (left to right, together with Multiplication)
A—Addition (left to right)
S—Subtraction (left to right, together with Addition)
1.4 Equations and Inequalities
Open Sentence—an equation with one or more variables
Ex. 17 – d = 14
Solution of an Equation—the number, in place of the variable, that makes an equation true
Ex. For 17 – d = 14, the solution to the equation is when d = 3
Inequality—when 2 expressions are joined by <, <, >, > (“larger” end “points” to larger number)
Solution of an Inequality—a number, in place of the variable, that makes an inequality true
Ex. For x + 3 > 7, the solution to the inequality includes x = 5, 6, 7, 8, etc.
1.5 A Problem Solving Plan Using Models
Word problems require translating words into math symbols. Words that trigger math symbols:
Addition: sum, and, more than, add, plus, increased by, add, etc.
Subtraction: less than, minus, difference, decreased by, subtract, etc.
Multiplication: product of, times, multiplied by, per, of, etc.
Division: quotient, divided by, fraction, etc.
Equals: totals, is, results in, gives, equals, etc.
Problem solving plan: label what is known (given) in the problem, translate words into math to
create a mathematical model, solve for unknown value
1.6 Tables and Graphs
Data—information, facts, or numbers that describe something
Bar Graph—tool to organize data with the length of each bar equaling the frequency of the data
Line Graph—tool to organize data that is measured in intervals of time. Ex. Stock market
Tables—a chart that provides the data to form graphs
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1.7 An Introduction to Functions
Function—a rule establishing a relationship between two quantities, inputs (x) and outputs (y)
A relationship is a function, if and only if, each input has exactly one output
Ex.
x | 2 4 2 5
is a function
x | 2 4 2 5 is not a function
y | 3 3 3 7
y | 3 3 4 7
Domain—the input values of a function (x)
Range—the output values of a function (y)
Chapter 2—Properties of Real Numbers
2.1 The Real Number Line
Origin—the point designated as “0” on a real number line
Positive Integers—the counting numbers to the “right” of the origin
Negative Integers—the counting numbers to the “left” of the origin
Opposites—two numbers that are the same distance from the origin (0), but on opposite sides of 0
Ex. -3 and 3
Ex. 3/7 and -3/7
Absolute Value—the positive distance between the origin (0) and the value inside | |
Ex. | -13 | = 13
Ex. | 43 | = 43
Ex. -| 5 | = -5
2.2 Addition of Real Numbers
Adding Numbers with the same sign: add the absolute values, use the common sign
Ex. -5 + -3 = | -5 | + | -3 | = 5 + 3 = 8, -8
Adding Numbers with different signs: subtract smaller absolute value from larger, use the sign of
the larger absolute value
Ex. -5 + 3 = | -5 | - | 3 | = 5 – 3 = 2, -2
Properties of Addition:
Commutative Property: a + b = b + a
Ex. 3 + 4 = 4 + 3
Associative Property: (a + b) + c = a + (b + c)
Ex. (2 + 3) + 4 = 2 + (3 + 4)
Identity Property: a + 0 = a
Ex. 5 + 0 = 5
Inverse Property: a + (-a) = 0
Ex. 4 + (-4) = 0
2.3 Subtraction of Real Numbers
Subtraction Rule: to subtract b from a, add the opposite of b to a.
Ex. 3 – 4 = 3 + (-4)
I.E. change subtraction problems into addition problems, then follow addition rules (2.2)
2.4 Adding and Subtracting Matrices
Not covered in Algebra 1
2.5 Multiplication of Real Numbers
Multiplication Rule: when multiplying two numbers with the same sign, the product is +; when
multiplying two numbers with different signs, the product is –
Multiplying More Than One Factor: Odd number of (–) factors, product will be -; Even number
of (–) factors, product will be +
Properties of Multiplication:
Commutative Property: a ∙ b = b ∙ a
Ex. 3 ∙ 4 = 4 ∙ 3
Associative Property: (a ∙ b) ∙ c = a ∙ (b ∙ c)
Ex. (3x)4 = 3(4x)
Identity Property: 1 ∙ a = a
Ex. 5/5 ∙ 7 = 7
Property of Zero: a ∙ 0 = 0
Ex. 329586 ∙ 0 = 0
Property of Opposites: a ∙ (-1) = -a
Ex. (-3)(-1) = 3
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2.6 The Distributive Property
Distributive Property:
a(b + c) = ab + ac
Ex. 5(x + 2) = 5x + 10
a(b – c) = ab – ac
Ex. 4(x – 7) = 4x – 28
Simplifying Expressions by Combining Like Terms:
Like Terms—terms in an expression that have the same variable raised to the same power
Ex. -8x and 35x
Ex. -4x4 and 10x4
Ex. 4 and 8
Coefficients—the constant multiplied by the variable
Ex. 7x, 7 is the coefficient
Constant—terms with no variables (plain old numbers)
Ex. 7
Simplified Expression—an expression without group symbols ( ) and all like terms
combined
Ex. 4(3x + 2x) = 12x + 8x = 20x
2.7 Division of Real Numbers
Reciprocal—the product of some number and its reciprocal is 1
Ex. -3 is the reciprocal of -1/3
Division Rule: to divide a number x by a non-zero number y, multiply x by the reciprocal of y
Ex. x ∕ y = x ∙ (1/y)
Ex. 3 / (1/2) = 3 ∙ 2 = 6
When dividing two numbers with the same sign, the quotient is +, when dividing two
numbers with different signs, the quotient is –
Dividing by Zero: undefined
Ex. 5 / 0 = undefined
Ex. 0 / 5 = 0
2.8 Probability and Odds
Probability—the likelihood of an event occurring—it’s a number (fraction) between 0 and 1
Theoretical Probability—number of favorable outcomes
total number of outcomes
Ex. Theoretical probability of tossing a “tail” when tossing a coin = ½
Experimental Probability—number of favorable outcomes observed
total number of trials
Ex. Toss a coin 7 times, 3 tails are observed, probability of observing tails = 3/7
Odds—number of favorable outcomes
number of unfavorable outcomes
Chapter 3—Solving Linear Equations
3.1 Solving Equations Using Addition and Subtraction
Solution of an Equation—the one number, in place of the variable, that makes the equation true
Isolating the Variable—finding an equivalent equation where x = the solution
Ex. 4x = 12, isolate x by dividing both sides of = by 4, x = 3
Inverse Operations—mathematical operations that “undo” each other (addition “undoes”
subtraction, etc.), that allow you to isolate the variable
Transforming Equations—using inverse operations to transform an original equation into an
equivalent equation where the variable is isolated and the solution to the equation is found
Ex. x + 7 = 12, x + 7 – 7 = 12 – 7, x = 5
Addition Property of Equality—adding the same number to both sides of an equation keeps the
equation true (used as an inverse operation to subtraction)
Ex. x – 5 = 9, x – 5 + 5 = 9 + 5, x = 14
Subtraction Property of Equality—subtracting the same number from both sides of an equation
keeps the equation true (used as an inverse operation to addition)
Ex. x + 8 = 11, x + 8 – 8 = 11 – 8, x = 3
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3.2 Solving Equations Using Multiplication and Division
Multiplication Property of Equality—multiplying the same number to both sides of an equation
keeps the equation true (used as an inverse operation to division)
Ex. x/5 = 10, (x/5) ∙ 5 = 10 ∙ 5, x = 50
Division Property of Equality—dividing both sides of an equation by the same number keeps the
equation true (used as an inverse operation to multiplication)
Ex. 6x = 30, 6x/6 = 30/6, x = 5
3.3 Solving Multi-Step Equations
Use multiple Inverse Operations to solve equations
Ex.
7x – 8 – 3x = 24
given equation
4x – 8
= 24
simplify by combining like terms
4x – 8 + 8 = 24 + 8
Addition Property of Equality
4x
= 32
simplify by combining like terms
4x/4
= 32/4
Division Property of Equality
x
=8
solve
3.4 Solving Equations with Variables on Both Sides
Combine like terms, then group like terms on the same side of (=) to solve equation:
Ex.
7x +
19 = -2x + 55
given equation
7x + 2x + 19 = -2x + 2x + 55 Addition Property of Equality
9x + 19
= 55
simplify by combining like terms
9x + 19 – 19 = 55 – 19
Subtraction Property of Equality
9x
= 36
simplify by combining like terms
9x/9
= 36/9
Division Property of Equality
x
= 4
solve
3.5 Linear Equations and Problem Solving
Linear Equation—an equation whose graph is a line (an equation containing a variable with an
exponent of 1)
Ex. y = 3x – 2
3.6 Solving Decimal Equations
Rounding Error—rounded solutions are not exactly correct and will have error
Solve decimal equations the same as normal equations, except solution may be rounded:
Ex.
17 = 48 – 6x
given equation
17 – 48 = 48 – 48 – 6x
Subtraction Property of Equality
-31 = -6x
simplify by combining like terms
-31/-6 = -6x/6
Division Property of Equality
-5.166666666 = x
solve
-5.1667 = x
round to 4 decimal places
3.7 Formulas and Functions
Literal Equations—an equations with multiple variables representing a real-world situation
Ex. C = 5/9(F – 32) to convert Fahrenheit to Celsius
Literal equations can solved for different variables
Ex. D = rt, solve for t, D/r = rt/r, D/r = t, so t = D/r
Functions are formulas or “rules” for equations (functions = equations)
Ex. y = 3x + 2 represents “y” as a function of “x”
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3.8 Rates, Ratios, and Percents
Ratio—a fraction, or a number expressed with a numerator and denominator
Ex. 4/5
Proportion—two equivalent ratios
Ex. ½ = 4/8
Ex. x/20 = 40/100
Rate (rate of “a” per “b”)—is a/b
Ex. rate of speed in miles (a) per hour (b)
Unit Rate—a rate per one given unit
Ex. rate: 300 miles in 3 hours, unit rate 100 miles in
1 hour
Solving Problems with Ratios: cross multiply, then solve for variable
Ex. x/20 = 40/100, 100x = 20(40), 100x = 800, x = 8
Percents—ratios using 100 as the denominator
To find percents, divide the number into the number of the whole amount
Ex. 7/35 = .20 = 20%
Chapter 4—Graphing Linear Equations and Functions
4.1 Coordinates and Scatter Plots
Coordinate Plane—formed by two number lines (x and y) intersecting at a right angle
Ordered Pair—each point on the Coordinate Plane, with an “address” of
(x-coordinate, y coordinate)
Ex. (4, -3)
Quadrants—the coordinate plane is broken into 4 sections:
y
quadrant II
quadrant I
quadrant III
x
quadrant IV
Scatter Plot—graph of a set of ordered pairs on the coordinate plane
4.2 Graphing Linear Equations
There are many techniques for graphing linear equations (equations whose graph is a line)
Plotting Points Method—select at least 2 values for x and solve for y, you now have at
least two ordered pairs, connect the ordered pairs with a straight line
Verifying Solutions of an Equations: to determine if a point (x, y) is a solution to an equation,
substitute the point into the equation for x and y and see if the equation is true—if true,
then point is the solution to the equation
Ex. Is (-3, 3) a solution to x + 3y = 6? (-3) + 3(3) = 6 -3 + 9 = 6 6 = 6 Yes
Horizontal Lines: y = b
where b is the y-intercept
Slope = 0
Ex. y = 7
Vertical Lines: x = a
where a is the x-intercept
Slope = undefined
Ex. x = -5
4.3 Quick Graphs Using Intercepts
x-intercept—the point where the line crosses the x (horizontal) axis; also the point where y = 0
y-intercept—the point where the line crosses the y (vertical) axis; also the point where x = 0
Intercept Method for Graphing Linear Equations: find the x-intercept and y-intercept, connect
with a straight line
Ex. Graph 2x + 3y = 6
Find x-intercept: 2x + 3(0) = 6, 2x + 0 = 6, 2x = 6, x = 3, x-int. = (3, 0)
Find y-intercept: 2(0) + 3y = 6, 0 + 3y = 6, 3y = 6, y = 2, y-int. = (0, 2)
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4.4 The Slope of a Line
Slope (m)—the number of units a line rises or falls for every unit of horizontal change—the ratio
of rise (vertical change) to run (horizontal change)
Slope between two points (x1, y1) and (x2, y2):
m = rise = change in y = (y2 – y1)
run change in x (x2 – x1)
Ex. (3, 4) and (7, 2) m = (2 – 4) = -2 = -1
(7 – 3) 4
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Rate of Change—a real world application of slope (slope = rate of change); compares two
different quantities that are changing.
Ex. velocity, (miles per hour) is a rate of change
4.5 Direct Variation
Direct Variation—when the increase or decrease of one unit of a variable (x) leads to the same rate
of increase or decrease in another variable (y)
Constant of Variation—the rate, k, that two variables (x and y) directly variate
Hint: The Constant of Variation (k) and the slope (m) are the same (m = k)
Ex. let x = number of Oreo cookies, y = number of chocolate cookie wafers, y = 2x
where 2 is the Constant of Variation
Model for Direct Variation: y = kx
Writing a Direct Variation Equation: substitute given x and y into Model of Direct Variation and
solve for k
Ex. x and y vary directly, when x = 3, y = 15, write a direct variation equation, then find
value of y when x = 10?
15 = k(3), k = 5, y = 5x is Model of Direct Variation
Now, when x = 10, y = 5(10), y = 50
4.6 Quick Graphs Using Slope Intercept
Slope-Intercept Form of a Linear Equation: y = mx + b where m = slope, b = y-intercept
Graphing Linear Equations in Slope-Intercept Form:
1.
Set up equation in Slope-Intercept form (y = mx + b) and identify slope and y-int.
2.
Graph the coordinates of the y-intercept (0, b)
3.
Change slope (m) into a fraction with numerator and denominator
4.
From the y-intercept, “rise” units of numerator and “run” units of denominator
5.
Connect y-intercept and second point with a straight line
Ex. y = 3/4x – 2
y-intercept at (0, -2), “rise” 3 units on y-axis, “run” 4 units on xaxis, connect (0, -2) and (4, 1) with a straight line
Parallel Lines—two lines are parallel ( // ) if they have the same slope
4.7 Solving Linear Equations Using Graphs
Not Covered in Algebra 1
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4.8 Functions and Relations
Relation—any set of ordered pairs
Function—a relation where each input value (x) has exactly one output value (y)
Ex.
x | 2 4 2 5
is a function
x | 2 4 2 5 is not a function
y | 3 3 3 7
y | 3 3 4 7
Vertical Line Test—a method to determine if a relation is a function—a relation is a function if
and only if any vertical line will pass through only 1 point of the graph
Ex
A function
Not a function
Function Notation—the symbol f(x) (“f of x”) replaces y in a linear function
Ex. equation: y = 3x – 5
and
function: f(x) = 3x – 5
are the same!!
Chapter 5—Writing Linear Equations
5.1 Writing Linear Equations in Slope-Intercept Form
Writing the equation in Slope-Intercept Form: substitute slope for m, y-intercept for b
Ex. m = -5/4 y-intercept = (0, 7)
y = -5/4x + 7
Writing the equation in Slope-Intercept Form from a graph: calculate slope between 2 points on
the graph using m = (y2 – y1)/ (x2 – x1), then identify y-intercept from graph, substitute into
y = mx + b
5.2 Writing Linear Equations Given the Slope and a Point
Writing the equation in Slope-Intercept Form when given a point (x, y) and the slope (m):
substitute the values for m, x, and y into y = mx + b, solve for b, use m and b for final
equation y = mx + b
Ex. point given (5, 4), m = 2:
4 = 2(5) + b, b = -6,
y = 2x – 6
Writing the equation in Slope-Intercept Form when given a point (x, y) and a line parallel:
Recall that parallel lines have the same slope (m), then follow steps from above
5.3 Writing Linear Equations Given Two Points
Writing the equation in Slope-Intercept Form when given two points (x1, y1) and (x2, y2):
Find the slope (m) between the 2 points using m = (y2 – y1)/ (x2 – x1), then select one of the
points and substitute the values for m, x, and y into y = mx + b, solve for b, use m and b for
the final equation y = mx + b
Ex. Given (3, 4) and (8, 7): m = (7 – 4)/(8 – 3) = 3/5, select (3, 4): 4 = 3/5(3) + b,
b = 11/5,
y = 3/5x + 11/5
Writing the Equation in Slope-Intercept Form when given 2 points (x1, y1) and (x2, y2) and a line
Perpendicular (┴):
Recall that perpendicular lines have slope m and -1/m (find the negative reciprocal of the
slope of the perpendicular line to find the slope of your line, then follow same steps
above)
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5.4 Fitting a Line to Data
Line of Best Fit—the line that best “fits” all of the ordered pairs in a scatter plot
Estimating the Equation of a Line of Best Fit: draw a line that best fits the ordered pairs, find the
slope (m) between 2 points on the line drawn using m = (y2 – y1)/ (x2 – x1), identify the yintercept by seeing where the line crosses the y-axis, use m and b to create y = mx + b
Correlation—the relationship between two variables (x and y)
••
• •
•
•
• •
• • •
• •
• ••
• •
•
•
• •
• •
Positive Correlation (+ m)
Negative Correlation (- m)
No Correlation
5.5 Point-Slope Form of a Linear Equation
Point-Slope Form of a Linear Equation—the equation of a non-vertical line (where slope is
undefined) that passes through a given point (x1, y1), with a given slope m is:
y – y1 = m(x – x1)
Ex. given (-3, 2) and m = 7: y – 2 = 7(x + 3)
Writing the equation in Point-Slope form when given 2 points only: find the slope using
m = (y2 – y1)/ (x2 – x1), then select one point to use with calculated m in Point Slope form
Ex. given (1, 3) and (9, 0): m = (0 -3)/(9 – 1) = -3/8
select (1, 3): y – 3 = -3/8(x – 1)
5.6 The Standard Form of a Linear Equation
Standard Form of a Linear Equation: Ax + By = C
where A and B are not both 0, and
coefficients A, B, and C are not written as fractions
Converting Slope-Intercept lines into Standard Form: “move” units of x and y to left of =
Ex. y = -4/5x + 10:
4/5x + y = 10, (now multiply every term by the denominator to remove fraction)
4/5x(5) + y(5) = 10(5),
4x + 5y = 50
Converting Point-Slope lines into Standard Form: create equation using Point-Slope formula,
simplify, then “move” units of x and y to left of =
Ex. given (4, 3) and m = 5: y – 3 = 5(x – 4),
y – 3 = 5x – 20,
-5x + y = -17 (almost standard form, multiply every term by -1)
5x – y = - 17
5.7 Predicting with Linear Models
Linear Interpolation—given a linear equation, using a value of x (within the original x values from
a set of ordered pairs) to solve for y
Linear Extrapolation—given a linear equation, using a value of x (outside the original x values
from a set of ordered pairs) to solve for y.
Ex. x | 5 4 7
linear equation (line of best fit): y = 0.9x – 1.6
y | 1 3 5
If x = 6, what is y (interpolation): y = 0.9(6) – 1.6,
y = 3.8
If x = 100 what is y (extrapolation): y = 0.9(100) – 1.6
y = 88.4