Download Solving One-Step Equations

ALGEBRA I NOTES
H. Bullard & L. Mills
Solving One-Step Equations
Multi-Step Equations w/ Distributive Property
Percent of Change
Relations
Midpoint of a Line
How To Graph a Line
Finding the Slope of a Line
Parallel and Perpendicular Lines
Polynomial Operations
Multiplying a Polynomial by a Monomial
end
2/8/2012
Solving One-Step Equations
4 +  = 10


+
?
=
Solving Multi-Step Equations
Multi-step Equations with Variables on Each Side
Solving Equations with Formulas
Linear Equations
Conversions Between Standard Form and SlopeIntercept Form
Line of Best Fit
Power to a Power
binomials times binomials
Algebra

    Concrete example

4 + x = 10
To solve:
Spring 2012
Algebra with a variable
4 + x = 10
-4
-4
0+x= 6
x= 6
(“=” means “same as”)
This means that 6 is the only number that can replace x to make the
equation true.
ALWAYS SHOW YOUR WORK!!
NO!
YES!
6=X+2
6=X+2
4=X
-2
-2
4=X
ASK YOURSELF: What is happening to the
variable? How can I get the variable by itself?
Examples: Addition/Subtraction
Notice: In these examples, you are moving an entire quantity (the constant in these examples) to the
other side of the equation.
1) 27 + n = 46
2) -5 + a = 21
-27
-27
Remember that whatever you
+5
+5
0 + n = 19
do on one side of the equation
you have to do on the other!
n = 19
a = 26
Examples: Multiplication/Division
Notice: In these examples, you are separating the quantity (the coefficient and the variable).
3) - 8n = -64
4) m = 12
Remember that whatever you
7
-8n = -64
do on one side of the equation
-8
-8
you have to do on the other!
7 * m = 12(7)
1 7
Page 1
H. Bullard & L. Mills
ALGEBRA I NOTES
Spring 2012
n=8
m = 84
5)
5b = 145
6)
5b = 145
5
5
-13 = m
-5
(-13)(-5) = m * -5
-5
65 = m
b = 29
Return to Top
2/9/2012
Solving Multi-Step Equations
Compare Equations
Equation
X + 3 = 15
X + 3 = 15
-3 -3
X = 12
Comments
 Total is the same 
Variable has a coefficient greater
than 1 
 One step
Equation 2
2x + 3 = 15
2x + 3 = 15
-3 = -3
2x
= 12
2x = 12
2
2
Two steps 
X=6
Coefficient = the number in front of a variable
Constant = a number. Example: in 2x + 3, 3 is the constant.
Like Terms = quantities with the same variable (eg 2x, 0.5x, 644x, 342909835x, 1/2x, etc, are like terms
because they all have an x). Numbers without a variable (constants) are also like terms!
How to Solve Multi-Step Equations:
1. Combine like terms on the left or right if you can.
2. Get all your variables on one side and all your constants on the other side (addition/subtraction OR
multiplication/division)
3. Divide both sides by the coefficient of the variable.
Examples:
A)
7 – y – y = -1
1. Combine like terms.
2. Get all variables on one side and all constants on
the other.
7 – 2y = -1
7 – 2y = -1
-7
-7
-2y = -8
3. Divide both sides by the coefficient of the
variable.
-2y = -8
-2
-2
Page 2
H. Bullard & L. Mills
ALGEBRA I NOTES
Spring 2012
y=4
B)
13 = 5 + 3b - 13
1. Combine like terms.
2. Get all variables on one side and all
constants on the other.
13 = 3b - 8
13 = 3b – 8
+8
+8
21 = 3b
3. Divide both sides by the coefficient of
the variable.
21 = 3b
3
3
7=b
C)
1. Combine like terms.
2. Get all variables on one side and all
constants on the other.
a – 18 = 2
5
z + 10 = 2
9
a – 18 = 2
5
+18 +18
a
= 20
5
9(z + 10) = 2(9)
9
z + 10 = 18
z + 10 = 18
-10 -10
z
=8
3. Divide both sides by the coefficient of
the variable.
a * 5 = 20 * 5
5
a = 120
Return to Top
2/10/2012
Solving Multi-Step Equations That Include the Distributive Property
Step 1: Distribute First
Step 2: Solve as you would any other multi-step
equation
Example: 2 = -2(n – 4)
2 = -2(n) + (-2)(-4)
2 = -2n + 8
- 8 = -2n – 8
- 6 = -2n
-6 = -2n
-2 -2
3=n
Page 3
ALGEBRA I NOTES
H. Bullard & L. Mills
-5(x + 3) = -45
-5x + (-15) = -45
+15 +15
-5x
= -30
x=6
Spring 2012
This means that 6 is the only number that you can substitute to make this equation
true.

Return to Top
2/14/2012
Solving Multi-Step Equations with Variables on Each Side
Step 1: Add like terms (there are none)
Step 2: Distribute (none to do)
Step 3: Variables on the left
Step 4: Numbers (constants) on the right
Example: 6 – x = 5x + 30
- 5x -5x
6 – 6x = 30
-6
-6
-6x = 24
-6 -6
x=4
Step 1: Add like terms (there are none)
Step 2: Distribute (none to do)
Step 3: Variables on the left
Step 4: Numbers (constants) on the right
Example: 5x + 2 = 2x - 10
- 2x
-2x
3x + 2 = - 10
-2
-2
3x = -12
3
3
x=4
Step 1: Add like terms
Step 2: Distribute (none to do)
Step 3: Variables on the left (if possible)
Step 4: Numbers (constants) on the right
Step 1: Add like terms (there are none)
Step 2: Distribute
Step 3: Variables on the left
Step 4: Numbers (constants) on the right
Example: 5y – 2y = 3y + 2
3y = 3y + 2
- 3y -3y
0=2
0 does not equal 2. It is a false statement. This
means that there are no real numbers that can be
substituted for y in this equation.
Example: 2y + 4 = 2(y + 2)
2y + 4 = 2(y) + 2(2)
2y + 4 = 2y + 4 
-2y
-2y
0+4=0+4

Both sides of the equation are equal without a
variable. Therefore, any real number can be
substituted for y, and the equation will be true.
Page 4
H. Bullard & L. Mills
ALGEBRA I NOTES
Spring 2012
Return to Top
2/16/12
Percent of Change
Types of INCREASES
New = original(1 + r)
* Tax (for example, on clothing purchases)
* Interest (such as on a bank account)
* inflation
* commission (such as when a car salesman earns a
certain % of what he sells)
Types of DECREASES
New = original(1-r)
* sales
* discounts
* depreciation
When solving a Percent of Change problem:
Step 1: WRITE OUT THE FORMULA!!!!
If you don’t it will be counted as wrong!
Step 2: Organize your information.
Step 3: Plug your values into the equation.
Step 4: Solve the equation.
EXAMPLE: original: 18
New = original(1-r)
new: 10
N =10
O = 18
r=?
10 = 18(1-r)
10 = 18 – 18r
-18 = -18
- 8 = - 18r
-18
-18
.4444 = r
44% = r
Return to Top
2/21/2012
Solving Equations with a Formula
Step 1: WRITE OUT THE FORMULA!!!!
Step 2: Organize your information.
Step 3: Plug your values into the equation.
Step 4: Solve the equation.
EXAMPLE: P = 2L + 2W
If the length is 5 and the perimeter is 24,
what is the value of W?
P = 2L + 2W
P = 24
L=5
W=?
24 = 2(5) + 2W
24 = 10 + 2W
-10 =-10 + 2W
14 = 2W
2
2
7=W
Page 5
ALGEBRA I NOTES
H. Bullard & L. Mills
Step 1: WRITE OUT THE FORMULA!!!!
Step 2: Organize your information.
Step 3: Plug your values into the equation.
Step 4: Solve the equation.
Spring 2012
EXAMPLE: P = 2L + 2W
If the width is 2.6 and the perimeter is
12.4, find L.
P = 2L + 2W
P = 12.4
L=?
W = 2.6
12.4 = 2L + 2(2.6)
12.4 = 2L + 5.2
- 5.2
-5.2
7.2 = 2L
2
2
3.6 = L
Return to the Top
RELATIONS
Domain = x values
Range = y values
f(x) = 2x2 + 7x – 2
f(2a) = 2(2a)2 + 7(2a) – 2
= 2(4a)2 + 14a – 2
= 8a2 + 14a – 2
Return to the top
Midpoint of a Line
The Midpoint of a Line is the middle, center, or halfway point of a line. Both segments of the line are
equal distances.
Midpoint = m x + x , y + y
2
2
(
)
Example: Find the midpoint between (2, -4) and (-10, -3)
1. Identify your x and y coordinates:
2. Put into the formula:
2 + (-10) ,
2
x y
(2, -4)
x y
(-10, -3)
-4 + (-3)
2
-8 , -7
2 2
(-4, -7/2) or (-4, -3.5) or (-4, -3½)
Page 6
H. Bullard & L. Mills
ALGEBRA I NOTES
Spring 2012
Example: Using the midpoint formula, we will find the missing information:
Given endpoint (-3, -7)
Midpoint (-8, 4)
Find endpoint (x, y)
Solving by formula:
x + x = xm , y + y = ym
2
2
2(-3 + x) = -8(2) , 2(-7 + y) = 4(2)
2
2
-3 + x = -16 , -7 + y = 8
+3
+3 +7
+7
x = - 13 ,
y = 15
(-13, 15)
Return to the Top
3/12/2012
Linear Equations
* What makes an equation a linear equation? Linear = makes a line
* What does linear look like?
1. No multiplication between variables together (in other words, no 2 variables are multiplied together)
2. No exponents greater than 1
A.
B.
C.
D.
Equation
2x = 3y + 1
Linear or Non-Linear
Linear
4xy + 2y = 7
2x2 = 4y - 3
X – 4y = 2
5 5
Non-linear
Non-linear
Linear
Linear equations can be written in standard form (Ax + By = C).
X-intercept –A point (x, y) where a line crosses the x-axis (x, 0)
Y-intercept – A point (x, y) where a line crosses the y-axis (0, y)
How do you graph a line?
1. Standard Form (Ax + By = C): use x and y intercepts
2. Slope-Intercept Form (y = mx + b): use graphing calculator:
y=
2nd graph
Plot
Page 7
Explanation
No multiplication between
variables; no exponents > 1
xy
x2
No multiplication between
variables; no exponents >1
H. Bullard & L. Mills
ALGEBRA I NOTES
Spring 2012
1. Graphing a line using Standard Form
Plot the x-intercept
Plot the y-intercept  find them by using the “Thumb Rule”
Example: Graph 2x + 5y = 20
1. Write your equation twice.
2. Cover (mark out) the x and its
coefficient in one; cover (mark
out) the y and its coefficient in
the other
3. Solve.
4. Write your ordered pairs.
5. Plot your points on a graph
and draw your line through
the points.
2x
X + 5y = 20
 This means x is 0,
so this ordered pair
will be (0, )
5y = 20
5
5
2x + 5y
X = 20
 This means y is 0,
so this ordered pair
will be ( , 0 )
2x = 20
2
2
y=4
x-intercept: (0, 4)
x = 10
y-intercept: (10, 0)
Example: Graph 3x + y = -1
1. Write your equation twice.
2. Cover (mark out) the x and its
coefficient in one; cover (mark
out) the y and its coefficient in
the other
3. Solve.
4. Write your ordered pairs.
5. Plot your points on a graph
and draw your line through
the points.
3x
X + y = -1
 This means x is 0,
so this ordered pair
will be (0, )
y = -1
3x +Xy = -1
 This means y is 0,
so this ordered pair
will be ( , 0 )
3x = -1
3
3
x-intercept: (0, -1)
x = -1/3
y-intercept: (-1/3, 0)
Example: Graph x – y = -3
1. Write your equation twice.
2. Cover (mark out) the x and its
coefficient in one; cover (mark
out) the y and its coefficient in
the other
3. Solve.
4. Write your ordered pairs.
5. Plot your points on a graph
and draw your line through
the points.
x - y = -3
X
 This means x is 0,
so this ordered pair
will be (0, )
-y = -3
You can’t have a –y, so multiply both
sides by -1:
-y(-1) = (-3)(-1)
y=3
x-intercept: (0, 3)
Page 8
x -X
y = -3
 This means y is 0,
so this ordered pair
will be ( , 0 )
x = -3
y-intercept: (-3, 0)
H. Bullard & L. Mills
ALGEBRA I NOTES
Spring 2012
2. Graphing a line using Slope-Intercept Form
y = mx + b
On your calculator, press:
Y=
2nd
graph (table)
Pick out 3 or 4 points from the table that are easy to plot (ie X and Y are whole numbers) and
plot them on a graph. Draw your line through those points.
Return to the top
3/14/12
Conversions Between Standard Form and Slope-Intercept Form
1. Converting from Standard Form to Slope-Intercept Form
Ax + By = C  y = mx + b
Example: Convert 4x + y = -2 to Slope-Intercept Form
1. Write your original formula.
4x + y = -2 – 4x
2. Get y by itself.
-4x
0
y = -4x -2
Example: Convert 5x – 3y = -6 to Slope-Intercept Form
1. Write your original formula.
5x – 3y = -6 – 5x
2. Get y by itself (move x).
-5x
0
-3y = -5x -6
3. Solve so that y is positive and
-3
-3 -3
has a coefficient of 1 (isolate y
and make it positive).
y = 5/3x + 2
NOTE: You are not separating the coefficient from the variable (the x
from the 4), so you are not dividing both sides by 4. You are moving the
entire quantity of 4x to the other side of the equation, which is why you
subtract. In other words, you are moving the variable, not isolating it.
NOTE: Write the sentence so that it follows the form. Therefore, you
write y = -4x – 2, rather than y = -2 – 4x.
NOTE: You are not separating the coefficient from the variable (the x
from the 5), so you are not dividing both sides by 5. You are moving the
entire quantity of 5x, which is why you subtract. In other words, you are
moving the variable, not isolating it.
NOTE: Write the sentence so that it follows the form. Therefore, you
write -3y = -5x – 6, rather than -3y = -6 – 5x.
NOTE: In this step you ARE separating the coefficient from the variable
(the -3 from the y), so you DO divide. In other words, you are isolating
the variable, not moving it.
NOTE: Write the slope (m) as a FRACTION, not a decimal.
Example: Convert 10x – y = 6 to Slope-Intercept Form
2. Write your original
10x – y = 6 – 10x
formula.
-10x
2. Get y by itself (move x).
0
(-1) -y = (-1)-10x + (-1)6
3. Solve so that y is positive and
has a coefficient of 1 (isolate y
and make it positive).
y = 10x + 6
Page 9
NOTE: You are not separating the coefficient from the variable (the x
from the 5), so you are not dividing both sides by 10. You are moving the
entire quantity of 10x, which is why you subtract. In other words, you are
moving the variable, not isolating it.
NOTE: Write the sentence so that it follows the form. Therefore, you
write -y = -10x – 6, rather than -y = -6 – 10x.
H. Bullard & L. Mills
ALGEBRA I NOTES
Spring 2012
2. Converting from Slope-Intercept Form to Standard Form
y = mx + b  Ax + By = C
To be standard form:
1. No fractions anywhere (includes decimals): all whole numbers
2. Leading x cannot be negative
3. Must be in simplest form
4x + 2y = 10  2x + y = 5
2 2
2
Example 1: Change y = 3/2x – 5 to Standard Form (Ax + By = C)
– 3/2x + y = (3/2x) – (3/2x) – 5
– 3/2x + y = -5
 This is negative and a fraction, so it is not yet in standard form.
(-2)(-3/2)x + (-2)y = -5(-2)
-3x -2y = 10
Example 2: Change y = 1/5x – 1 to Standard Form (Ax + By = C)
– 1/5x + y = (1/5x) – (1/5x) – 1
– 1/5x + y = -1
 This is negative and a fraction, so it is not yet in standard form.
(-5)(-1/5)x + (-5)y = -1(-5)
x -5y = 10
Return to the top
Finding the Slope of a Line
Slope is rate of change
 It is the steepness of a line
 Represented by a fraction
m = slope = rise =  y = change in y
run  x change in x
m=4=4
1
Positive slope
Negative slope
Zero slope (m = 0)
negative
positive
Undefined (no slope)
Page 10
H. Bullard & L. Mills
ALGEBRA I NOTES
Spring 2012
Two ways to find slope:
1. graph
2. formula
m=y–y
x–x
Example: Find the slope of the line going through the points (-4, 4) and (4, 6)
m=y–y
m=y–y
x–x
x–x
4 – 6 = -2 = 1
-4 – 4 -8 4
6-4=2=1
4-(-4) 8 4
Example: Find the slope of the line going through the points (-4, -1) and (-2, -5)
m = y – y = -1 – (-5) = 4 = -2
x – x -4 – (-2) -2
Return to the top
3/16/12
PARALLEL AND PERPENDICULAR LINES
Parallel Lines (||)
 Do not intersect
 Have the same slopes
Example: Write the slope-intercept form of an equation of a line parallel to y = 4x – 2 and passing through
the point (-2, 2)
1. Use the formula
y – y1 = m(x – x1)
y – y1 = m(x – x1)
2. Organize your information
m=4
x1 = -2
y1 = 2
3. Plug your values into your formula.
y – 2 = 4(x – (-2))
4. Solve
y – 2 = 4(x + 2)
y – 2 = 4x + 8
+2
+2
y = 4x + 10
Perpendicular Lines ()
 Intersect
 Slopes are 1) opposite (change the sign)
2) reciprocal (flip them)
Page 11
H. Bullard & L. Mills
ALGEBRA I NOTES
Spring 2012
Example: Write the slope-intercept form of the  line y = 1/2x + 1, crossing through point (4, 2).
1. Use the formula
y – y1 = m(x – x1)
y – y1 = m(x – x1)
2. Convert your slope
Opp
flip
m = ½  - ½  - 2/1 = -2
3. Organize your information
m = -2
x1 = 4
y1 = 2
4. Plug your values into your formula.
y – 2 = -2(x – 4)
5. Solve
y – 2 = -2x + 8
+2
+2
y = -2x + 10
Example: Write the slope-intercept form of the  line 2x + 4y = 12, crossing through point (-1, 3).
1. Change the form of the original line
2x + 4y = 12 - 2x
from standard form to slope-intercept
-2x
form.
4y = -2x – 12
4
4
1. Use the formula
y – y1 = m(x – x1)
2. Convert your slope
3. Organize your information
4. Plug your values into your formula.
5. Solve
y = -1/2x - 3
y – y1 = m(x – x1)
Opp flip
m = -½  ½  2/1 = 2
m=2
x1 = -1
y1 = 3
y – 3 = 2(x – (-1))
y – 3 = 2x + 2
+3
+3
y = 2x + 5
Return to the top
3/19/12
Line of Best Fit
Inside a SCATTERPLOT are data (ordered pairs). The more data you have, the better your line will be.
Why do we need a line? TO PREDICT! (for example, it helps business owners predict profits, whether to
expand or downsize, etc).
Page 12
ALGEBRA I NOTES
H. Bullard & L. Mills
Line of Best Fit
Regression Equation
Prediction Equation
Best Fit Line
Spring 2012
 Equation of a line to predict.
X (L1)
Y (L2)
Independent Variable
Dependent Variable
** See “Best-Fit Line” Handout**
Types of Questions you can be asked with Line of Best Fit
1. To find the equation
2. To make a prediction (from a line)
3. What does slope mean/represent?
4. What does y-intercept represent?
Example: The cab driver charges a $5 flat fee and $.25 per mile.
y = .25x + 5
y represents the total bill
.25 represents the charge or RATE
x represents per mile
5 represents the flat fee or starting point; the total bill starting at 0 miles
Return to the Top
4/9/12
Polynomial Operations
Monomial: one term
A number, a variable, or the product of 1 or more variables
Examples: 5, a, 6x, 5x2yz
Binomial: two terms
Addition or subtraction of two monomials
Examples: 5-5x; 2x2+3
Trinomial: three terms
Addition or subtraction of three monomials
Example: 6+7x2-3x
Adding Polynomials:
1) (4a - 5) + (3a + 6)
Combine like terms/regroup
(4a - 5) + (3a + 6)
4a + 3a - 5 + 6
7a + 1
Page 13
H. Bullard & L. Mills
ALGEBRA I NOTES
2) (6xy + 2y + 6x) + (4xy – x)
10xy + 2y + 5x
Subtracting Polynomials:
1)
DISTRIBUTE the -1 on the 2nd monomial
Combine like terms
(3a – 5) – (5a + 1)
(3a – 5) + (- 5a – 1)
- 2a - 6
2)
DISTRIBUTE the -1 on the 2nd monomial
Combine like terms
(9xy + y – 2x) – (6xy – 2x)
(9xy + y – 2x) +(- 6xy + 2x)
3xy + y + 0
3xy + y
Return to top
4/10/12
Multiplying Monomials
1. Multiply coefficient (whole number)
2. Add exponents
What does y3 mean? y * y * y
What does (x2)(x4) mean? x * x |* x * x * x * x  x6
x2+4 = x6
EXAMPLES:
1) (x16) (x163) = x16+163 = x179
2) (4x3y2) (-2xy)
(4) (-2) x3+1 y2+1
-8 x4 y3
3) (-5xy) (4x2) (y4)
(-5) (4) (1) x3+1 y2+1
4) (-3j2k4) (2jk6)
(-3) (2) j2+1 k4+6
-6j3k10
Power to a Power
(x2)3 = x2 * x2 * x2 = x2+2+2 = x6
xx xx xx
= x6
2*3
6
x = x
RULE: Multiply exponents.
EXAMPLES:
1) (xy3)4 = xy3 * xy3 * xy3 * xy3 = x4 x12
= x1*4 y3*4 = x4 x12
Page 14
Spring 2012
ALGEBRA I NOTES
H. Bullard & L. Mills
4/11/12
Multiplying a Polynomial by a Monomial
RULE: Distribute!
EXAMPLES:
1) x(5x + x2)
x(5x) + x(x2)
5x2 + x3
2) -2xy(2xy + 4 x2)
(-2xy) (2y) + (-2xy) (4 x2)
-4x y2 + -8 x3y
3) 2x2y2(3xy + 2y + 5x)
6x3y3 + 4x2y3 + 10x3y2
Return to top
4/12/12
Binomials Times Binomials
There are 3 techniques you can use for multiplying polynomials:
1. Distributive Property
2. FOIL Method
3. The Box Method
EXAMPLE OF DISTRIBUTIVE PROPERTY OR FOIL METHOD:
(2x + 3) (5x + 8)
First: (2x)(5x) = 10x2
Outer: (2x)(8) = 16x
Inner: (3)(5x) = 15x
Last: (3)(8) = 24
Combine like terms: 10x2 + 31x + 24
EXAMPLES OF BOX METHOD:
(3x – 5) (5x + 2)
3x
-5
2
5x 15 x
-25x
15 x2 – 25x + 6x – 10
2
6x
-10
15 x2 – 19x – 10
(2x – 5) (x2 – 5x + 4)
x2
-5x
2x
2 x3
-10 x2
-5
-5x2
25x
4
8x
-20
2x3 - 10x2 - 5x2 + 8x + 25x - 20
2x3 - 15x2 + 33x - 20
Page 15
Spring 2012
H. Bullard & L. Mills
ALGEBRA I NOTES
4/13/12
Dividing Monomials
When dividing monomials, subtract the exponents
1. b5 = b * b * b * b * b = b5-2 = b3
b2
b*b
end
Page 16
Spring 2012