Download Class Notes 10-5-2010

Chapter 2.3
Further solving linear equations
Learning objectives
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Apply the General Strategy for solving linear equations
Solve equations containing fractions and decimals
Recognize identities and equations with no solution
Vocabulary: identity; no solution
A general strategy for solving linear
equations
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If the equation contains fractions, multiply BOTH sides by
the LCD to clear the equation of fractions
Use the distributive property to remove the parenthesis
Combine like terms on each side of the equation
Isolate all variable terms on one side of the equation and
the number terms on the other side using addition
Get the variable alone using the multiplicative property
Check the solution by plugging it into the original
equation
Solve 4(2x – 3) + 7 = 3x + 5
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1. there are no fractions, proceed to step 2
2. distribute 4(2x – 3) to 4(2x) – 4(3) = 8x – 12
You now have 8x – 12 + 7 = 3x + 5
3. combine like terms: 8x + (-12 + 7) = 8x – 5
You now have 8x – 5 = 3x + 5
4. Get all variable terms on 1 side: subtract 3x from both
You now have 8x – 3x – 5 = 3x – 3x + 5;
That leaves you with 5x – 5 = 5.
5. Get the variable alone by adding 5 to both. 5x – 5 + 5 = 5 +
5
You now have 5x = 10. Divide each side by 5.
Finally, you have x = 2. Substitute 2 into each x to check.
Solve 8(2 – t) = -5t
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Use the distributive property:
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Add 8t to both sides:
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Combine like terms:
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Divide both sides by 3.
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Check your work:
Solve the following linear equations
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6a – (5a – 1) = 4
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4(3b – 1) = 16
Solve the following linear equations
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4z = 8(2z + 9)
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2(x + 8) = 3(x – 5)
Solve the following linear equations
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3(2a – 3) = 5(a + 4)
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12(4c – 2) = 3c - 4
Solving equations containing fractions or
decimals
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Solve x/2 – 1 = 2x/3 – 3
First, we clear away fractions by multiplying by LCD of 2
and 3, which is 6.
6(x/2 – 1) = 6(2x/3 – 3)
3x – 6 = 4x – 18
Subtract 3x from each side; then add 18 to each side
3x – 3x – 6 = 4x – 3x -18; -6 = x – 18
-6 + 18 = x – 18 + 18
12 = x ; now check your work
12/2 – 1 = 6 – 1 = 5; 2(12)/3 – 3 = 24/3 – 3 = 8 – 3 = 5;
Helpful hint: page 110
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When solving an equation, it makes no difference on
which side of the equation the variable terms lie; they can
lie on the left or the right; just as long as the number
terms lie on the other side.
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12 = x is the same as saying x = 12
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Many money problems include decimal calculations.
Solve 0.25x + 0.10(x – 3) = 1.1
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Clear decimals by multiplying each side by 100. This
moves decimal points two spaces to the right.
25x + 10(x – 3) = 110
Now distribute: 25x + 10x – 30 = 110
Combine like terms: 35x – 30 = 110
Add 30 to both sides: 35x – 30 + 30 = 110 + 30
35x = 140
Divide both sides by 35
35x/35 = 140/35; x = 4
Check work: 0.25(4) + 0.10(4-3) = 1 + 0.10(1) = 1.1
Solve each equation containing fractions
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y/6 – 4 = 1
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¼ x – 3/8 x = 5
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(-6x + 5)/4 + 1 = -5x/4
Solve each equation containing decimals
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0.05x + 0.06(x – 1500) = 570
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0.4(x + 7) – 0.1(3x + 6) = -0.8
Recognizing identities and equations with
no solution
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So far, every equation we have worked on has a solution.
But how about: x + 5 = x + 7 ??
In the “equation” above, x represents the same real
number. But what number satisfies the conditions?
Simplify the equation: subtract 5 from each; x = x + 2?
Then subtract x from each side: 0 = 2?
This is FALSE, and thus this “equation” is not an equation
at all. It has no solution.
Now, how about x + 6 = x + 6?
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Subtract 6 from each side; now x = x
The variable x can be replaced by ANY number and it will
always be true.
This type of equation is called an identity. It is always true,
no matter what you set x to
Solve: -2(x – 5) + 10 = -3(x + 2) + x
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Distribute: -2x + 10 + 10 = -3x – 6 + x
Combine like terms: -2x + 20 = -2x – 6
Add 2x to both sides: -2x + 2x + 20 = -2x + 2x - 6
The result: 20 = -6?
This is false; therefore, there is no solution to this equation
Solve 3(x – 4) = 3x - 12
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Distribute: 3x – 12 = 3x – 12
We seem to have the same thing on both sides of the
equation. Add 12 to both sides: 3x – 12 + 12 = 3x – 12 +
12
So 3x = 3x? Any real value of x makes this true.
This equation is an identity
Solve each equation.
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6(z + 7) = 6z + 42
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3 + 12x – 1 = 8x + 4x – 1
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x/3 – 3 = 2x/6 + 1
Chapter 2.4: An introduction to problem
solving
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We have been problem solving all along
This section deals with translating spoken or written
problems into numerical expressions to be solved
First: understand the problem. Choose a variable to
represent the unknown quantity
Second: translate the problem into an equation
Then: solve the equation
Finally, interpret the results. Check the solution in the
original equation, and state your conclusion
Finding an unknown number
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Twice the sum of a number and 4 is the same as four
times the number, decreased by 12.
If we let x be the unknown number, then
“the sum of a number and 4” becomes x + 4;
 “twice the sum of a number and 4” becomes 2(x + 4)
 “four times the number” becomes 4x
 “four times the number decreased by 12”: 4x – 12
So the translation becomes: 2(x + 4) = 4x – 12
Solve for x; 2x + 8 = 4x – 12; 2x = 4x – 20; -2x = -20;
The equation is solved with x = 10
Check our work: 2(10 + 4) = 4(10) – 12; 28 = 28
Conclusion: the number is 10.
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A 10 foot board is to be cut into 2 pieces so
that the longer piece is 4 times the shorter.
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Let x be the length of the shorter piece.
The length of the longer piece will therefore be: 4x
We also know the two pieces together will = 10
Translate the problem: x + 4x = 10
Solve: x + 4x = 5x = 10; divide each side by 5
Solution: x = 2
Interpret: the shorter piece of board will be 2 feet, and
the longer will be 4 x 2 feet = 8 feet. 2 ft + 8 ft = 10 ft.
State: the shorter piece of board will be 2 feet. The
longer piece shall be 8 feet.
A student has a PT job with Geek Squad. He charges
$20/hr to come to your home, and $25/hr labor.
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Last month he made 10 calls and made $575. How many
hours did he spend working on computers?
Understand: let h be the number of labor hours.
He earned 10($20) = $200 for just showing up
He earns $25 for each labor hour; he made $575 total
Translate: 10($20) + h($25) = $575
Distribute: $200 + $25h = $575
Subtract 200 from each side: $25h = $575 - $200 = $375
Divide each side by $25: h = 15
Interpret: the student charged 15 hours labor last month.
Solve these problems
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Eight is added to a number and the sum is double, and the
result is -11 less than the number.
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The difference between two positive integers is 42. One
integer is three times as great as the other.
Solve these problems
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The room numbers of two adjacent rooms are two
consecutive odd numbers. Their sum is 1380.
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The left and right pages of a book are consecutive
integers. The sum of the two is 349. What is the 1st page?
Solve these problems
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A graduating class has 450 students. There are 206 more
girls than boys. How many boys are there?
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A 22 foot pipe is cut into two pieces. The shorter piece
is 7 feet shorter than the longer. What is the length of
the longer piece?