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QuestionSimplify the following expressions:
1. (6r+7)+(13+7r)(6 r+7)+(13+7 r)
2. (7r−32)−(23+6r)\left(7 r-\frac{3}{2}\right)-\left(\frac{2}{3}+6 r\right)
3. (13−32r)−(1−r)\left(13-\frac{3}{2} r\right)-(1-r)
4. (−8−r)+(2r−4)(-8-r)+(2 r-4)

Studdy Solution

STEP 1

Assumptions1. We are given four expressions to simplify. . We assume that the operations of addition and subtraction are performed from left to right.
3. We assume that the variable rr can be any real number.

STEP 2

Let's start with the first expression. We simplify it by combining like terms.
(6r+7)+(13+7r)(6 r+7)+(13+7 r)

STEP 3

Rearrange the terms so that the like terms are together.
=6r+7r+7+13=6r +7r +7 +13

STEP 4

Combine the like terms.
=13r+20=13r +20

STEP 5

Now, let's move on to the second expression. We simplify it by combining like terms.
(7r−32)−(23+r)\left(7 r-\frac{3}{2}\right)-\left(\frac{2}{3}+ r\right)

STEP 6

istribute the negative sign to the terms in the second parentheses.
=r−32−23−6r=r - \frac{3}{2} - \frac{2}{3} -6r

STEP 7

Rearrange the terms so that the like terms are together.
=7r−6r−32−23=7r -6r - \frac{3}{2} - \frac{2}{3}

STEP 8

Combine the like terms.
=r−32−23= r - \frac{3}{2} - \frac{2}{3}

STEP 9

Combine the fractions by finding a common denominator.
=r−96−46= r - \frac{9}{6} - \frac{4}{6}

STEP 10

Combine the fractions.
=r−136= r - \frac{13}{6}

STEP 11

Now, let's move on to the third expression. We simplify it by combining like terms.
(13−3r)−(−r)\left(13-\frac{3}{} r\right)-(-r)

STEP 12

istribute the negative sign to the terms in the second parentheses.
=−2r−+r= - \frac{}{2}r - + r

STEP 13

Rearrange the terms so that the like terms are together.
=13−−32r+r=13 - - \frac{3}{2}r + r

STEP 14

Combine the like terms.
=12−32r+r=12 - \frac{3}{2}r + r

STEP 15

Combine the terms with rr by converting rr to 22r\frac{2}{2}r.
=12−32r+22r=12 - \frac{3}{2}r + \frac{2}{2}r

STEP 16

Combine the terms with rr.
=12−2r=12 - \frac{}{2}r

STEP 17

Finally, let's move on to the fourth expression. We simplify it by combining like terms.
(−−r)+(2r−4)(--r)+(2 r-4)

STEP 18

Rearrange the terms so that the like terms are together.
=−8−r+2r−4= -8 - r +2r -4

STEP 19

Combine the like terms.
=−8−4−r+r= -8 -4 - r +r

STEP 20

Combine the like terms.
=−12+r= -12 + rSo, the simplified forms of the given expressions are(6r+7)+(13+7r)=13r+20(6 r+7)+(13+7 r) =13r +20(7r−3)−(3+6r)=r−136\left(7 r-\frac{3}{}\right)-\left(\frac{}{3}+6 r\right) = r - \frac{13}{6} (13−3r)−(−r)=12−r\left(13-\frac{3}{} r\right)-(-r) =12 - \frac{}{}r(−8−r)+(r−4)=−12+r(-8-r)+( r-4) = -12 + r

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