Solving Linear Equations - §2.1 - 2.2
Fall 2013 - Math 1010
Expressions vs. Equations
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Roadmap
I
Solve linear equations.
I
Evaluate linear equations.
I
Solve the real-world problems: percents, ratios, proportions.
I
Solve the real-world problems: business models, mixtures, rates.
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§2.1 - Equations
Equations are statements equating two algebraic expressions.
For equations with variables, solving that equations is a search to find a
value for the variable for which the equation is true.
This can end in failure. (Some have no solutions.)
This can end in a successful search. (Some have conditional solutions.)
This can end in a hugh success. (When any, all, every real number solves
the equation, it is called an identity.)
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§2.1 - Seriously
Seriously. Check your work by evaluating.
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§2.1 - Example:
Find any solutions to the equation. Then check your solution.
−3x − 5 = 4x + 16
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§2.1 - Example:
Find any solutions to the equation. Then check your solution.
−3x − 5 = 4x + 16
Which steps in the work did you show?
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§2.1 - Example:
Find any solutions to the equation. Then check your solution.
−3x − 5 = 4x + 16
Which steps in the work did you show?
Did you check your solution?
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§2.1 - Example:
Find any solutions to the equation. Then check your solution.
1
2x − 4 = (6x − 3)
3
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§2.1 - Example:
Find any solutions to the equation. Then check your solution.
1
2x − 4 = (6x − 3)
3
Which steps in the work did you show?
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§2.1 - Example:
Find any solutions to the equation. Then check your solution.
1
2x − 4 = (6x − 3)
3
Which steps in the work did you show?
Did you check your answer?
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§2.2 - Problem Solving
The goal of §2.2 is to translate a problem into an equation and solve the
equation. Often times, we already have methods to solve problems.
Dispite this, using the recently discussed skills is our task.
§2.2 offers problems on percents, ratios, and proportions. *Optional §2.3
offers problems in business, mixing, and the classical rate problems. I
believe §2.3 is very useful, though questions like these will not be in
homework, quizzes, or exams.
Tip: How to use a ’rate:’ Comparing two numbers (in different
measurements of units) is done with a fraction called a rate. Can any
number (like 78) be a rate? Sure! Just write it as 78
1 .
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§2.2 - Example
Write an algebraic equation for the following problem. Follow step from
§1.5 to set up a mathematical model from the verbal statements.
A squirrel has stored 7 acorns. It stores 6 acorns per week. The squirrel
needs 55 acorns stored. How many months will it take to reach that goal?
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§2.2 - Example
Write an algebraic equation for the following problem. Follow step from
§1.5 to set up a mathematical model from the verbal statements.
A squirrel has stored 7 acorns. It stores 6 acorns per week. The squirrel
needs 55 acorns stored. How many months will it take to reach that goal?
I choose four weeks per month. It follows that the monthly rate is 24
acorns each month.
Verbal model: (Survival goal) = 24 × Time + (Current amount.)
Labels: Survival goal = 55, Time = t, Current amount = 7.
Equation: 55 = 24t + 7.
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§2.2 - Percents
A good model for percent problems to use:
(Compared number) = (Percent in decimal form) · (Base number)
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§2.2 - Percents
A good model for percent problems to use:
(Compared number) = (Percent in decimal form) · (Base number)
Examples Each statement uses a variable for an unknown value.
1) The number 8.7 is what percent of 9.8?
2) What number is 7.2% of 128?
3) 0.112 is 81.3% of what number?
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§2.2 - Ratios
Percents compare one number to 100. This is one example of a ratio.
Ratios can compare any two numbers.
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§2.2 - Ratios
Percents compare one number to 100. This is one example of a ratio.
Ratios can compare any two numbers.
Examples Each statement wants a ratio of two (different) measurements
in the same unit.
Tip: Many of these require knowledge of conversion factors.
1) Find the ratio of 30 miles per hour to 114 kilometers per hour.
2) Find the ratio of brocolli crowns priced at $1.99 per pound to brocolli
priced at $1.49 per pound.
3) Which is better to buy, a 12-inch cake at $5.59, or a 14-inch cake at
$6.29?
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§2.2 - Proportions
Percents are ratios of a number and 100, the last examples deal with
general ratios. The third example in the slide on ratios compares two
ratios. When two ratios are equal, they are proportional.
Is the third example from the last slide a proportion?
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§2.2 - Proportions
Percents are ratios of a number and 100, the last examples deal with
general ratios. The third example in the slide on ratios compares two
ratios. When two ratios are equal, they are proportional.
Is the third example from the last slide a proportion?
Examples Solve each proportion.
1) A 12 oz can of soda costs 85 cents. How much will a proportional 20
oz bottle cost?
2)
y −5
5
=
15
12
3) The property tax on a $45,000 valued home is $5,000. Under the same
conditions what is the value of a home when taxed $8,200?
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Assignment
Assignment:
For Wednesday:
1. Read sections 2.4, 2.5.
2. Exercises from §2.1, 2.2 due Wednesday, September 11.
3. Quiz #2 Wednesday, September 11.
Vocabulary: equation, identity, condition solution, equivalence, linear
equation, modelling, base numbers, percent, ratio, proportion
Understand: The difference in language for problems with just
expressions, and problems with just equations. Solving a linear equation.
Checking a solution to an equation. Mathematically modelling real-world
problems like percent, ratio, and proportion problems.
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