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QuestionSimplify the expression 6y−24y2−23y+1y2\frac{6 y-2}{4 y^{2}}-\frac{2}{3 y}+\frac{1}{y^{2}} and choose the correct result. Options: a. y+33y2\frac{y+3}{3 y^{2}}, b. 10y+610 y+6, c. 10y−512y2\frac{10 y-5}{12 y^{2}}, d. 5y+36y2\frac{5 y+3}{6 y^{2}}.

Studdy Solution

STEP 1

Assumptions1. The expression is 6y−4y−3y+1y\frac{6 y-}{4 y^{}}-\frac{}{3 y}+\frac{1}{y^{}} . We need to simplify this expression

STEP 2

First, we need to find a common denominator for all the terms in the expression. The common denominator of 4y24y^2, yy, and y2y^2 is 12y212y^2.

STEP 3

Now, rewrite each term with the common denominator.6y−2y2=3(6y−2)3(y2)=18y−612y2\frac{6 y-2}{ y^{2}} = \frac{3(6 y-2)}{3( y^{2})} = \frac{18 y-6}{12 y^{2}}−23y=−(2)(3y)=−812y2-\frac{2}{3 y} = -\frac{(2)}{(3 y)} = -\frac{8}{12 y^{2}}1y2=12(1)12(y2)=1212y2\frac{1}{y^{2}} = \frac{12(1)}{12(y^{2})} = \frac{12}{12 y^{2}}

STEP 4

Substitute these equivalent terms back into the original expression.
18y−612y2−812y2+1212y2\frac{18 y-6}{12 y^{2}} - \frac{8}{12 y^{2}} + \frac{12}{12 y^{2}}

STEP 5

Since all the terms have the same denominator, we can combine the numerators.
18y−−8+1212y2\frac{18 y- -8 +12}{12 y^{2}}

STEP 6

implify the numerator.
18y−212y2\frac{18 y -2}{12 y^{2}}

STEP 7

Factor out the common factor of2 from the numerator.
2(9y−1)12y2\frac{2(9 y -1)}{12 y^{2}}

STEP 8

implify the fraction by cancelling out the common factor of2.
y−16y2\frac{ y -1}{6 y^{2}}This matches option c. 10y−512y2\frac{10 y-5}{12 y^{2}}.

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