bell work: solve for x: 5y + x – 2y – 4 + 3x = 0
TRANSCRIPT
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Bell Work:
Solve for x:
5y + x – 2y – 4 + 3x = 0
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Answer:
x = -3/4 y + 1
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Lesson 43:Least Common Multiple, Least
Common Multiples of Algebraic
Expressions
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If we are given the numbers
4, 5, and 8
And are asked to find the smallest number that is evenly divisible by each of the numbers, a reasonable guess would be the product of the numbers, which is 160, because we know that each of the numbers will divide 160 evenly
160/4 = 40 160/5 = 32 160/8 = 20
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But 160 is not the smallest number that is evenly divisible by the three numbers. The number 40 is.
40/4 = 10 40/5 = 8 40/8 = 5
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We call the smallest number that can be divided evenly by each of a group of specified numbers the least common multiple (LCM) of the specified numbers.
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We can find the LCM of some numbers by making mental calculations, but it is nice to have a special procedure to use if some of the numbers are large numbers. The procedure is as follows:
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1. Write each number as a product of prime factors.
2. Compute the LCM by using every factor of the given numbers as a factor of the LCM. Use each factor the greatest number of times it is a factor in any of the numbers.
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To demonstrate this procedure we will find the LCM of
18, 81, and 500
First we write each number as a product of prime factors:
18 = 2 x 3 x 3
81 = 3 x 3 x 3 x 3
500 = 2 x 2 x 5 x 5 x 5
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Now we find the LCM by using the procedure in Step 2. The number 2 is a factor of both 18 and 500. it appears twice in 500, so it will appear twice in the LCM.
2 x 2
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The number 3 is a factor of both 18 and 81. it appears four times in 81, so it will appear four times in the LCM.
2 x 2 x 3 x 3 x 3 x 3
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Therefore, 40,500 is the smallest number that is evenly divisible by each of the three numbers 18, 81, and 500.
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Example:
Find the LCM of 8, 15, and 100.
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Answer:
8 = 2 x 2 x 2
15 = 3 x 5
100 = 2 x 2 x 5 x 5
2 x 2 x 2 x 3 x 5 x 5
= 600
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Example:
Find the Least Common Multiple of 30, 75, and 80.
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Answer:
30 = 2 x 3 x 5
75 = 3 x 5 x 5
80 = 2 x 2 x 2 x 2 x 5
2 x 2 x 2 x 2 x 3 x 5 x 5
= 1200
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Practice:
Find the LCM of 560, 588, and 1250.
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Answer:
560 = 2 x 2 x 2 x 2 x 5 x 7
588 = 2 x 2 x 3 x 7 x 7
1250 = 2 x 5 x 5 x 5 x 5
2 x 2 x 2 x 2 x 3 x 5 x 5 x 5 x 5 x 7 x 7
= 1, 470, 000
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The least common multiple is most often encountered when it is used as the least common denominator. If we are asked to add the fractions
¼ + 5/8 + 7/12
We rewrite each of these fractions as a fraction whose denominator is 24, which is the least common multiple of 4, 8 and 12.
6/24 + 15/24 + 14/24 = 35/24
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In lesson 44 we will discuss the method of adding algebraic fractions. To prepare for that, we will practice finding the least common multiple of algebraic expressions.
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Example:
Find the least common multiple of 15a b and 10ab .2 3
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Answer:
15a b = 3 x 5 x a x a x b
10ab = 2 x 5 x a x b x b x b
LCM = 2 x 3 x 5 x a x a x b x b x b
= 30a b
2
3
2 3
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Practice:
Find the LCM of 4x m and 6x m.
2 3
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Answer:
4x m = 2 2 x x m
6x m = 2 3 x x x m
LCM = 2 2 3 x x x m
= 12x m
2
3
3
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Practice:
Find the LCM of 12x am and 14x am .
2 2
3 4
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Answer:
84x am3 4
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HW: Lesson 43 #1-30