Solved on Mar 07, 2024

Find 2 equations with the same $x$ value as $\frac{2}{3}(6 x+12)=-24$ . Options: $4 x+8=-24$ , $9 x+18=-24$ , $4 x=-16$ , $\frac{18 x+36}{2}=-24$ , $4 x=-32$ .

STEP 1

Assumptions
1. We are given the equation $\frac{2}{3}(6x+12)=-24$ .
2. We need to find which of the given options have the same solution for $x$ .
3. The options are: a. $4x+8=-24$ b. $9x+18=-24$ c. $4x=-16$ d. $\frac{18x+36}{2}=-24$ e. $4x=-32$

STEP 2

First, we will solve the given equation for $x$ .
$\frac{2}{3}(6x+12)=-24$

STEP 3

Multiply both sides of the equation by $\frac{3}{2}$ to eliminate the fraction.
$\left(\frac{3}{2}\right)\left(\frac{2}{3}\right)(6x+12)=\left(\frac{3}{2}\right)(-24)$

STEP 4

Simplify both sides of the equation.
$6x+12=-36$

STEP 5

Subtract 12 from both sides of the equation to isolate the term with $x$ .
$6x+12-12=-36-12$

STEP 6

Simplify the equation further.
$6x=-48$

STEP 7

Divide both sides of the equation by 6 to solve for $x$ .
$\frac{6x}{6}=\frac{-48}{6}$

STEP 8

Calculate the value of $x$ .
$x=-8$

STEP 9

Now that we have the value of $x$ , we will check which of the given options have the same solution for $x$ by substituting $x=-8$ into each option.

STEP 10

Substitute $x=-8$ into option a.
$4(-8)+8=-24$

STEP 11

Calculate the left-hand side of option a.
$-32+8=-24$

STEP 12

Check if the equation holds true.
$-24=-24$

STEP 13

Since the equation holds true, option a has the same value of $x$ .

STEP 14

Substitute $x=-8$ into option b.
$9(-8)+18=-24$

STEP 15

Calculate the left-hand side of option b.
$-72+18=-54$

STEP 16

Check if the equation holds true.
$-54 \neq -24$

STEP 17

Since the equation does not hold true, option b does not have the same value of $x$ .

STEP 18

Substitute $x=-8$ into option c.
$4(-8)=-16$

STEP 19

Calculate the left-hand side of option c.
$-32=-16$

STEP 20

Check if the equation holds true.
$-32 \neq -16$

STEP 21

Since the equation does not hold true, option c does not have the same value of $x$ .

STEP 22

Substitute $x=-8$ into option d.
$\frac{18(-8)+36}{2}=-24$

STEP 23

Calculate the left-hand side of option d.
$\frac{-144+36}{2}=\frac{-108}{2}$

STEP 24

Calculate the left-hand side of option d further.
$-54=-24$

STEP 25

Check if the equation holds true.
$-54 \neq -24$

STEP 26

Since the equation does not hold true, option d does not have the same value of $x$ .

STEP 27

Substitute $x=-8$ into option e.
$4(-8)=-32$

STEP 28

Calculate the left-hand side of option e.
$-32=-32$

STEP 29

Check if the equation holds true.
$-32=-32$

STEP 30

Since the equation holds true, option e has the same value of $x$ .
The equations that have the same value of $x$ as $\frac{2}{3}(6x+12)=-24$ are option a: $4x+8=-24$ and option e: $4x=-32$ .

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Find 2 equations with the same xxx value as 23(6x+12)=−24\frac{2}{3}(6 x+12)=-2432​(6x+12)=−24. Options: 4x+8=−244 x+8=-244x+8=−24, 9x+18=−249 x+18=-249x+18=−24, 4x=−164 x=-164x=−16, 18x+362=−24\frac{18 x+36}{2}=-24218x+36​=−24, 4x=−324 x=-324x=−32.

STEP 1

STEP 2

First, we will solve the given equation for xxx.23(6x+12)=−24\frac{2}{3}(6x+12)=-2432​(6x+12)=−24

STEP 3

Multiply both sides of the equation by 32\frac{3}{2}23​ to eliminate the fraction.(32)(23)(6x+12)=(32)(−24)\left(\frac{3}{2}\right)\left(\frac{2}{3}\right)(6x+12)=\left(\frac{3}{2}\right)(-24)(23​)(32​)(6x+12)=(23​)(−24)

STEP 4

Simplify both sides of the equation.6x+12=−366x+12=-366x+12=−36

STEP 5

Subtract 12 from both sides of the equation to isolate the term with xxx.6x+12−12=−36−126x+12-12=-36-126x+12−12=−36−12

STEP 6

Simplify the equation further.6x=−486x=-486x=−48

STEP 7

Divide both sides of the equation by 6 to solve for xxx.6x6=−486\frac{6x}{6}=\frac{-48}{6}66x​=6−48​

STEP 8

Calculate the value of xxx.x=−8x=-8x=−8

STEP 9

Now that we have the value of xxx, we will check which of the given options have the same solution for xxx by substituting x=−8x=-8x=−8 into each option.

STEP 10

Substitute x=−8x=-8x=−8 into option a.4(−8)+8=−244(-8)+8=-244(−8)+8=−24

STEP 11

Calculate the left-hand side of option a.−32+8=−24-32+8=-24−32+8=−24

STEP 12

Check if the equation holds true.−24=−24-24=-24−24=−24

STEP 13

Since the equation holds true, option a has the same value of xxx.

STEP 14

Substitute x=−8x=-8x=−8 into option b.9(−8)+18=−249(-8)+18=-249(−8)+18=−24

STEP 15

Calculate the left-hand side of option b.−72+18=−54-72+18=-54−72+18=−54

STEP 16

Check if the equation holds true.−54≠−24-54 \neq -24−54=−24

STEP 17

Since the equation does not hold true, option b does not have the same value of xxx.

STEP 18

Substitute x=−8x=-8x=−8 into option c.4(−8)=−164(-8)=-164(−8)=−16

STEP 19

Calculate the left-hand side of option c.−32=−16-32=-16−32=−16

STEP 20

Check if the equation holds true.−32≠−16-32 \neq -16−32=−16

STEP 21

Since the equation does not hold true, option c does not have the same value of xxx.

STEP 22

Substitute x=−8x=-8x=−8 into option d.18(−8)+362=−24\frac{18(-8)+36}{2}=-24218(−8)+36​=−24

STEP 23

Calculate the left-hand side of option d.−144+362=−1082\frac{-144+36}{2}=\frac{-108}{2}2−144+36​=2−108​

STEP 24

Calculate the left-hand side of option d further.−54=−24-54=-24−54=−24

STEP 25

Check if the equation holds true.−54≠−24-54 \neq -24−54=−24

STEP 26

Since the equation does not hold true, option d does not have the same value of xxx.

STEP 27

Substitute x=−8x=-8x=−8 into option e.4(−8)=−324(-8)=-324(−8)=−32

STEP 28

Calculate the left-hand side of option e.−32=−32-32=-32−32=−32

STEP 29

Check if the equation holds true.−32=−32-32=-32−32=−32

STEP 30

Since the equation holds true, option e has the same value of xxx.The equations that have the same value of xxx as 23(6x+12)=−24\frac{2}{3}(6x+12)=-2432​(6x+12)=−24 are option a: 4x+8=−244x+8=-244x+8=−24 and option e: 4x=−324x=-324x=−32.

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Find 2 equations with the same $x$ value as $\frac{2}{3}(6 x+12)=-24$ . Options: $4 x+8=-24$ , $9 x+18=-24$ , $4 x=-16$ , $\frac{18 x+36}{2}=-24$ , $4 x=-32$ .

First, we will solve the given equation for $x$ .
$\frac{2}{3}(6x+12)=-24$

Multiply both sides of the equation by $\frac{3}{2}$ to eliminate the fraction.
$\left(\frac{3}{2}\right)\left(\frac{2}{3}\right)(6x+12)=\left(\frac{3}{2}\right)(-24)$

Simplify both sides of the equation.
$6x+12=-36$

Subtract 12 from both sides of the equation to isolate the term with $x$ .
$6x+12-12=-36-12$

Simplify the equation further.
$6x=-48$

Divide both sides of the equation by 6 to solve for $x$ .
$\frac{6x}{6}=\frac{-48}{6}$

Calculate the value of $x$ .
$x=-8$

Now that we have the value of $x$ , we will check which of the given options have the same solution for $x$ by substituting $x=-8$ into each option.

Substitute $x=-8$ into option a.
$4(-8)+8=-24$

Calculate the left-hand side of option a.
$-32+8=-24$

Check if the equation holds true.
$-24=-24$

Since the equation holds true, option a has the same value of $x$ .

Substitute $x=-8$ into option b.
$9(-8)+18=-24$

Calculate the left-hand side of option b.
$-72+18=-54$

Check if the equation holds true.
$-54 \neq -24$

Since the equation does not hold true, option b does not have the same value of $x$ .

Substitute $x=-8$ into option c.
$4(-8)=-16$

Calculate the left-hand side of option c.
$-32=-16$

Check if the equation holds true.
$-32 \neq -16$

Since the equation does not hold true, option c does not have the same value of $x$ .

Substitute $x=-8$ into option d.
$\frac{18(-8)+36}{2}=-24$

Calculate the left-hand side of option d.
$\frac{-144+36}{2}=\frac{-108}{2}$

Calculate the left-hand side of option d further.
$-54=-24$

Check if the equation holds true.
$-54 \neq -24$

Since the equation does not hold true, option d does not have the same value of $x$ .

Substitute $x=-8$ into option e.
$4(-8)=-32$

Calculate the left-hand side of option e.
$-32=-32$

Check if the equation holds true.
$-32=-32$

Since the equation holds true, option e has the same value of $x$ .
The equations that have the same value of $x$ as $\frac{2}{3}(6x+12)=-24$ are option a: $4x+8=-24$ and option e: $4x=-32$ .