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QuestionDivide the rational expressions and simplify: 3x−615÷5x−103\frac{3 x-6}{15} \div \frac{5 x-10}{3}.

Studdy Solution

STEP 1

Assumptions1. We are given two rational expressions, 3x−615\frac{3x-6}{15} and 5x−103\frac{5x-10}{3}. . We are asked to divide these two expressions.
3. The result should be simplified to its lowest terms.

STEP 2

To divide two fractions, we multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by swapping its numerator and denominator.
ab÷cd=ab×dc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

STEP 3

Apply the rule from2 to our problem.
3x−615÷5x−103=3x−615×35x−10\frac{3x-6}{15} \div \frac{5x-10}{3} = \frac{3x-6}{15} \times \frac{3}{5x-10}

STEP 4

When multiplying fractions, we simply multiply the numerators together to get the new numerator, and the denominators together to get the new denominator.
ab×cd=a×cb×d\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}

STEP 5

Apply the rule from4 to our problem.
3x−15×35x−10=(3x−)×315×(5x−10)\frac{3x-}{15} \times \frac{3}{5x-10} = \frac{(3x-) \times3}{15 \times (5x-10)}

STEP 6

implify the multiplication in the numerator and denominator.
(3x−6)×315×(5x−10)=9x−1875x−150\frac{(3x-6) \times3}{15 \times (5x-10)} = \frac{9x-18}{75x-150}

STEP 7

We can simplify the fraction by factoring out common factors in the numerator and the denominator.

STEP 8

Factor out from the numerator and75 from the denominator.
x−1875x−150=(x−2)75(x−2)\frac{x-18}{75x-150} = \frac{(x-2)}{75( x-2)}

STEP 9

Now, we can cancel out the common factor (x−2)(x-2) from the numerator and the denominator.
9(x−2)75(x−2)=975\frac{9(x-2)}{75( x-2)} = \frac{9}{75}

STEP 10

The fraction 975\frac{9}{75} can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is3.
975=9/375/3=325\frac{9}{75} = \frac{9/3}{75/3} = \frac{3}{25} So, the result of the division of the given rational expressions, in its simplest form, is 325\frac{3}{25}.

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