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3 years ago

Solve: 1. 1/6y− 1/2 =3− 1/2y 2. 4x+1/15 = 2x/10

Mathematics

1 answer:

kogti [31]3 years ago

5 0

Answer:

First Equation → y = 21/4

Second Equation → x = -1/57

Explanation:

solving equation #1

step 1 - simplify

$\displaystyle\frac{1}{6}y - \displaystyle\frac{1}{2} = 3 - \displaystyle\frac{1}{2}y\\\\\displaystyle\frac{1}{6}* \displaystyle\frac{y}{1} - \displaystyle\frac{1}{2} = 3 - \displaystyle\frac{1}{2}* \displaystyle\frac{y}{1}\\\\\displaystyle\frac{y}{6} - \displaystyle\frac{1}{2} = 3 - \displaystyle\frac{y}{2}$

step 3 - multiply each side of the equation by six

$\displaystyle\frac{y}{6} - \displaystyle\frac{1}{2} = 3 - \displaystyle\frac{y}{2}\\\\\displaystyle\frac{y}{6} * \displaystyle\frac{6}{1}- \displaystyle\frac{1}{2} * \displaystyle\frac{6}{1}= \displaystyle\frac{3}{1} *\displaystyle\frac{6}{1} - \displaystyle\frac{y}{2}* \displaystyle\frac{6}{1}\\\\y - 3 = 18 - 3y$

step 4 - add three to both sides of the equation.

$y - 3 = 18 - 3y\\\\y-3+3=18+3-3y\\\\y = -3y+21$

step 5 - add three y to both sides of the equation.

$y = -3y+21\\\\y+3y = -3y+3y+21\\\\y+3y=21$

step 6 - simplify

$y+3y=21\\\\4y=21$

step 7 - divide both sides of the equation by four

$4y=21\\\\\displaystyle\frac{4y}{4} = \displaystyle\frac{21}{4}\\ \\y = \displaystyle\frac{21}{4}$

Therefore, the solution to the first given equation is y = 21/4 or y = 5.25.

solving equation #2

step 1 - simplify.

$4x + \displaystyle\frac{1}{15} = \displaystyle\frac{2x}{10} \\\\4x + \displaystyle\frac{1}{15} = 2*\displaystyle\frac{x}{10}\\\\4x+\displaystyle\frac{1}{15} = \displaystyle\frac{x}{5}$

step 2 - multiply each side of the equation by five.

$4x+\displaystyle\frac{1}{15} = \displaystyle\frac{x}{5}\\\\\displaystyle\frac{4x}{1}* \displaystyle\frac{5}{1} +\displaystyle\frac{1}{15} * \displaystyle\frac{5}{1} = \displaystyle\frac{x}{5}* \displaystyle\frac{5}{1} \\\\20x + \displaystyle\frac{1}{3} = x$

step 3 - subtract twenty x from each side of the equation.

$20x + \displaystyle\frac{1}{3} = x \\\\20x -20x+ \displaystyle\frac{1}{3} = x -20x\\\\\displaystyle\frac{1}{3} = -19x$

step 4 - divide each side of the equation by negative nineteen.

$\displaystyle\frac{1}{3} = -19x\\\\\displaystyle\frac{\displaystyle\frac{1}{3} }{-19} = \displaystyle\frac{-19x}{-19} \\\\-\displaystyle\frac{1}{57} = x$

step 5 - switch

$-\displaystyle\frac{1}{57} =x\\\\x = -\displaystyle\frac{1}{57}$

Therefore, the solution to the second equation is x = -1/57.

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Note, if the expected frequencies for all the variables are calculated corrrectly then ∑Ei= n, in this case

75.6+31.5+18.9= 126 ⇒ Is a quick way to check if the calculatios are well done, especially when you are working with more categories.

c) The conditions for this test are:

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If p-value ≥ α, the decision is to reject the null hypothesis.

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Using a level of significance of 5%, there is no significant evidence to reject the null hypothesis. Then you can conclude that the book format the students choose to use for the introductory statistics course follows the theoretical percentages expected by the professor.

I hope this helps!

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