2.1 the addition property of equality a linear equation in one variable can be written in the form:...
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2.1 The Addition Property of Equality
• A linear equation in one variable can be written in the form: Ax + B = 0
• Linear equations are solved by getting “x” by itself on one side of the equation
• Addition Property of Equality: if A=B then A+C=B+C
• General rule: Whatever you do to one side of the equation, you have to do the same thing to the other side.
2.1 The Addition Property of Equality
• Example of Addition Property:
x – 5 = 12 (add 5 to both sides)x = 17
• Example using subtraction:
kk
kkkk
kk
55
53
58
53
53
58
53
17
17
17
2.2 The Multiplication Property of Equality
• Multiplication Property of Equality: if A=B and C is non-zero, then AC=BC (both equations have the same solution set)
• Since division is the same as multiplying by the reciprocal, you can also divide each side by a number.
• General rule: Whatever you do to one side of the equation, you have to do the same thing to the other side.
2.2 The Multiplication Property of Equality
• Example of Multiplication Property:
• Example using division
5x = 60 (divide both sides by 5)x = 12
8
6)(
6
324
16
34
34
43
34
43
h
h
h
2.3 More on Solving Linear Equations: Terms - Review
• As with expressions, a mathematical equation is split up into terms by the +/-/= sign:
• Remember, if the +/- sign is in parenthesis, it doesn’t count:
33 61
21
32 xxx
361
21
32 xxx
2.3 More on Solving Linear Equations: Multiplying Both Sides by a Number
• Multiply each term by the number (using the distributive property).
• Within each term, multiply only one factor.
Notice that the y+1 does not become 4y+4
48124
12434
xx
xx
481124
124134
xyx
xyx
2.3 More on Solving Linear Equations: Clearing Fractions
• Multiply both sides by the Least Common Denominator (in this case the LCD = 4):
17
125
1225
342
45
4
4
32
15
4
1
x
x
xx
xx
xx
2.3 More on Solving Linear Equations: Clearing Decimals
• Multiply both sides by the smallest power of 10 that gets rid of all the decimals
16805
30505
3055010305510
3.10005.10051.100
3.05.51.
xx
x
xxxx
xx
xx
2.3 More on Solving Linear Equations: Why Clear Fractions?
• It makes the calculation simpler:
104
9410
941029452
19494
945
47
94
194
15
47
1
x
x
xxxx
xx
xx
2.3 More on Solving Linear Equations
• 1 – Multiply on both sides to get rid of fractions• 2 – Use the distributive property to remove
parenthesis• 3 – Combine like terms• 4 – Put variables on one side, numbers on the
other by adding/subtracting on both sides• 5 – Get “x” by itself on one side by multiplying
both sides• 6 – Check your answers
2.3 More on Solving Linear Equations
• Example:36
121
32 xxx
1834 xxx
187 xx186 x3x
2.4 An Introduction to Applications for Linear Equations
• 1 – Decide what you are asked to find• 2 – Write down any other pertinent information
(use other variables, draw figures or diagrams )• 3 – Translate the problem into an equation.• 4 – Solve the equation.• 5 – Answer the question posed.• 6 – Check the solution.
2.4 An Introduction to Applications for Linear Equations
• Find the measure of an angle whose complement is 10 larger.
1. x is the degree measure of the angle.2. 90 – x is the degree measure of its complement3. 90 – x = 10 + x4. Subtract 10: 80 – x = x
Add x: 80 = 2xDivide by 2: x = 40
5. The measure of the angle is 40 6. Check: 90 – 40 = 10 + 40
2.5 Formulas - examples
• A = lw• I = prt• P = a + b + c• d = rt• V = LWH• C = 2r
• Area of rectangle• Interest• Perimeter of triangle• Distance formula• Volume – rectangular solid• Circumference of circle
2.5 Formulas
• Example: d=rt; (d = 252, r = 45)
then 252 = 45tdivide both sides by 45:
45275t
2.6 Ratios and Proportions
• Ratio – quotient of two quantities with the same units
Examples: a to b, a:b, or
Note: percents are ratios where the second number is always 100:
ba
10035%35
2.6 Ratios and Proportions
• Proportion – statement that two ratios are equal
Examples:
Cross multiplication:
if then
dc
ba
bcad dc
ba
2.6 Ratios and Proportions
• Solve for x:
Cross multiplication:
so x = 63
7981 x
x9567x 9781
2.7 More about Problem Solving
• Percents are ratios where the second number is always 100:
Example:
If 70% of the marbles in a bag containing 40 marbles are red, how many of the marbles are red?:
#red marbles =
73.%73 10073
284010070
2.7 More about Problem Solving
• How many gallons of a 12% indicator solution must be mixed with a 20% indicator solution to get 10 gallons of a 14% solution? Let x= #gallons of 12% solution,then 10-x= #gallons of 20% solution :
)10%(14)10%(20)%(10 xx4.10.2)%(20)%(10 xx
6.)%(10 x61.
6. x
2.8 Solving Linear Inequalities
• < means “is less than” means “is less than or equal to”
• > means “is greater than” means “is greater than or equal to”
note: the symbol always points to the smaller number
2.8 Solving Linear Inequalities
• A linear inequality in one variable can be written in the form:
ax < b (a0)
• Addition property of inequality:
if a < b then a + c < b + c
2.8 Solving Linear Inequalities
• Multiplication property of inequality:– If c > 0 then
a < b and ac < bc are equivalent
– If c < 0 thena < b and ac > bc are equivalent
note: the sign of the inequality is reversed when multiplying both sides by a negative number
2.8 Solving Linear Inequalities
• Example:36
121
32 xxx
1834 xxx
18 xx182 x
9x-9