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Zorluk: OrtaSolving Linear Equations

A botanist models the growth of a rare seedling. The number of weeks ww that the seedling has been growing satisfies the linear equation:

14(3w−8)+0.6=15(2w+7)\frac{1}{4}(3w - 8) + 0.6 = \frac{1}{5}(2w + 7)

If the seedling's growth continues to follow this model, what is the value of 3 less than 5 times the number of weeks the seedling has been growing?

  1. A
    -37
  2. B
    -17
  3. C
    22
  4. D
    25
  5. 37Cevap

Cevap

37
Solving the given equation for ww yields w=8w = 8. The question asks for the value of 3 less than 5 times the number of weeks, which translates to the expression 5w−35w - 3. Substituting w=8w = 8 into the expression results in 5(8)−3=375(8) - 3 = 37.

Adım Adım Çözüm

1
Multiply both sides of the equation by the least common multiple of the denominators, which is 20.
5(3w−8)+12=4(2w+7)5(3w - 8) + 12 = 4(2w + 7)
This clears the fractions and simplifies the equation to integer coefficients.
2
Distribute the coefficients across the terms inside the parentheses.
15w−40+12=8w+2815w - 40 + 12 = 8w + 28
This removes the parentheses so terms can be grouped.
3
Combine the constant terms on the left side of the equation.
15w−28=8w+2815w - 28 = 8w + 28
Combining −40-40 and 1212 simplifies the expression on the left.
4
Isolate the variable ww by moving the variable terms to the left side and constant terms to the right side.
7w=567w = 56
Subtracting 8w8w and adding 2828 to both sides groups like terms together.
5
Solve for ww by dividing both sides by 7.
w=8w = 8
This isolates the variable ww to find the number of weeks.
6
Translate '3 less than 5 times the number of weeks' into an algebraic expression and evaluate it for w=8w = 8.
5w−3=5(8)−3=375w - 3 = 5(8) - 3 = 37
This translates the verbal question into mathematical terms and calculates the final value.

Anahtar Kavram

Solving multi-step linear equations containing fractions and decimals, and translating verbal expressions into algebraic terms.

Alternatif Yöntem

Instead of clearing the fractions first, one could convert the fractions to decimals: 0.25(3w−8)+0.6=0.2(2w+7)0.25(3w - 8) + 0.6 = 0.2(2w + 7). Distribute to get 0.75w−2+0.6=0.4w+1.40.75w - 2 + 0.6 = 0.4w + 1.4, which simplifies to 0.75w−1.4=0.4w+1.40.75w - 1.4 = 0.4w + 1.4. Subtracting 0.4w0.4w and adding 1.41.4 to both sides yields 0.35w=2.80.35w = 2.8, which gives w=2.80.35=8w = \frac{2.8}{0.35} = 8. Then, compute 5(8)−3=375(8) - 3 = 37.
Tahmini Süre:1m 30s
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Solving Linear Equations

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