Question Find the equation with a unique solution from the given options: (1) $9x+1=9x+11$ , (2) $4x+8=4(x+2)$ , (3) $5x+2=3x+14$ , (4) $6(x-1)=2(3x+5)$ .

Studdy Solution

STEP 1

Assumptions
1. We are looking for an equation with exactly one solution.
2. We have four options to choose from.
3. An equation with exactly one solution will have different coefficients for the variable on both sides of the equation.

STEP 2

Analyze option (1): $9x + 1 = 9x + 11$

STEP 3

Subtract $9x$ from both sides of the equation to see if we can isolate $x$ .
$9x + 1 - 9x = 9x + 11 - 9x$

STEP 4

Simplify both sides of the equation.
$1 = 11$

STEP 5

Since $1 \neq 11$ , this equation has no solution. Therefore, option (1) is not the correct choice.

STEP 6

Analyze option (2): $4x + 8 = 4(x + 2)$

STEP 7

Distribute the $4$ on the right side of the equation.
$4x + 8 = 4x + 8$

STEP 8

Subtract $4x$ from both sides of the equation.
$4x + 8 - 4x = 4x + 8 - 4x$

STEP 9

Simplify both sides of the equation.
$8 = 8$

STEP 10

Since the equation simplifies to a true statement and does not involve the variable $x$ , this equation has infinitely many solutions. Therefore, option (2) is not the correct choice.

STEP 11

Analyze option (3): $5x + 2 = 3x + 14$

STEP 12

Subtract $3x$ from both sides of the equation to isolate terms with $x$ on one side.
$5x + 2 - 3x = 3x + 14 - 3x$

STEP 13

Simplify both sides of the equation.
$2x + 2 = 14$

STEP 14

Subtract $2$ from both sides of the equation to isolate $2x$ .
$2x + 2 - 2 = 14 - 2$

STEP 15

Simplify both sides of the equation.
$2x = 12$

STEP 16

Divide both sides by $2$ to solve for $x$ .
$\frac{2x}{2} = \frac{12}{2}$

STEP 17

Simplify both sides of the equation.
$x = 6$

STEP 18

Since we found a single value for $x$ , option (3) has exactly one solution. Therefore, option (3) is a potential correct choice. We should still check option (4) to confirm.

STEP 19

Analyze option (4): $6(x - 1) = 2(3x + 5)$

STEP 20

Distribute the $6$ on the left side and the $2$ on the right side of the equation.
$6x - 6 = 6x + 10$

STEP 21

Subtract $6x$ from both sides of the equation.
$6x - 6 - 6x = 6x + 10 - 6x$

STEP 22

Simplify both sides of the equation.
$-6 = 10$

STEP 23

Since $-6 \neq 10$ , this equation has no solution. Therefore, option (4) is not the correct choice.

STEP 24

After analyzing all options, we have determined that option (3) is the only equation with exactly one solution.
The equation with exactly one solution is: $5x + 2 = 3x + 14$

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Question Find the equation with a unique solution from the given options: (1) 9x+1=9x+119x+1=9x+119x+1=9x+11, (2) 4x+8=4(x+2)4x+8=4(x+2)4x+8=4(x+2), (3) 5x+2=3x+145x+2=3x+145x+2=3x+14, (4) 6(x−1)=2(3x+5)6(x-1)=2(3x+5)6(x−1)=2(3x+5).

Studdy Solution

STEP 1

Assumptions1. We are looking for an equation with exactly one solution.2. We have four options to choose from.3. An equation with exactly one solution will have different coefficients for the variable on both sides of the equation.

STEP 2

Analyze option (1): 9x+1=9x+119x + 1 = 9x + 119x+1=9x+11

STEP 3

Subtract 9x9x9x from both sides of the equation to see if we can isolate xxx.9x+1−9x=9x+11−9x9x + 1 - 9x = 9x + 11 - 9x9x+1−9x=9x+11−9x

STEP 4

Simplify both sides of the equation.1=111 = 111=11

STEP 5

Since 1≠111 \neq 111=11, this equation has no solution. Therefore, option (1) is not the correct choice.

STEP 6

Analyze option (2): 4x+8=4(x+2)4x + 8 = 4(x + 2)4x+8=4(x+2)

STEP 7

Distribute the 444 on the right side of the equation.4x+8=4x+84x + 8 = 4x + 84x+8=4x+8

STEP 8

Subtract 4x4x4x from both sides of the equation.4x+8−4x=4x+8−4x4x + 8 - 4x = 4x + 8 - 4x4x+8−4x=4x+8−4x

STEP 9

Simplify both sides of the equation.8=88 = 88=8

STEP 10

Since the equation simplifies to a true statement and does not involve the variable xxx, this equation has infinitely many solutions. Therefore, option (2) is not the correct choice.

STEP 11

Analyze option (3): 5x+2=3x+145x + 2 = 3x + 145x+2=3x+14

STEP 12

Subtract 3x3x3x from both sides of the equation to isolate terms with xxx on one side.5x+2−3x=3x+14−3x5x + 2 - 3x = 3x + 14 - 3x5x+2−3x=3x+14−3x

STEP 13

Simplify both sides of the equation.2x+2=142x + 2 = 142x+2=14

STEP 14

Subtract 222 from both sides of the equation to isolate 2x2x2x.2x+2−2=14−22x + 2 - 2 = 14 - 22x+2−2=14−2

STEP 15

Simplify both sides of the equation.2x=122x = 122x=12

STEP 16

Divide both sides by 222 to solve for xxx.2x2=122\frac{2x}{2} = \frac{12}{2}22x​=212​

STEP 17

Simplify both sides of the equation.x=6x = 6x=6

STEP 18

Since we found a single value for xxx, option (3) has exactly one solution. Therefore, option (3) is a potential correct choice. We should still check option (4) to confirm.

STEP 19

Analyze option (4): 6(x−1)=2(3x+5)6(x - 1) = 2(3x + 5)6(x−1)=2(3x+5)

STEP 20

Distribute the 666 on the left side and the 222 on the right side of the equation.6x−6=6x+106x - 6 = 6x + 106x−6=6x+10

STEP 21

Subtract 6x6x6x from both sides of the equation.6x−6−6x=6x+10−6x6x - 6 - 6x = 6x + 10 - 6x6x−6−6x=6x+10−6x

STEP 22

Simplify both sides of the equation.−6=10-6 = 10−6=10

STEP 23

Since −6≠10-6 \neq 10−6=10, this equation has no solution. Therefore, option (4) is not the correct choice.

STEP 24

After analyzing all options, we have determined that option (3) is the only equation with exactly one solution.The equation with exactly one solution is: 5x+2=3x+145x + 2 = 3x + 145x+2=3x+14

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Question Find the equation with a unique solution from the given options: (1) $9x+1=9x+11$ , (2) $4x+8=4(x+2)$ , (3) $5x+2=3x+14$ , (4) $6(x-1)=2(3x+5)$ .

Assumptions
1. We are looking for an equation with exactly one solution.
2. We have four options to choose from.
3. An equation with exactly one solution will have different coefficients for the variable on both sides of the equation.

Analyze option (1): $9x + 1 = 9x + 11$

Subtract $9x$ from both sides of the equation to see if we can isolate $x$ .
$9x + 1 - 9x = 9x + 11 - 9x$

Simplify both sides of the equation.
$1 = 11$

Since $1 \neq 11$ , this equation has no solution. Therefore, option (1) is not the correct choice.

Analyze option (2): $4x + 8 = 4(x + 2)$

Distribute the $4$ on the right side of the equation.
$4x + 8 = 4x + 8$

Subtract $4x$ from both sides of the equation.
$4x + 8 - 4x = 4x + 8 - 4x$

Simplify both sides of the equation.
$8 = 8$

Since the equation simplifies to a true statement and does not involve the variable $x$ , this equation has infinitely many solutions. Therefore, option (2) is not the correct choice.

Analyze option (3): $5x + 2 = 3x + 14$

Subtract $3x$ from both sides of the equation to isolate terms with $x$ on one side.
$5x + 2 - 3x = 3x + 14 - 3x$

Simplify both sides of the equation.
$2x + 2 = 14$

Subtract $2$ from both sides of the equation to isolate $2x$ .
$2x + 2 - 2 = 14 - 2$

Simplify both sides of the equation.
$2x = 12$

Divide both sides by $2$ to solve for $x$ .
$\frac{2x}{2} = \frac{12}{2}$

Simplify both sides of the equation.
$x = 6$

Since we found a single value for $x$ , option (3) has exactly one solution. Therefore, option (3) is a potential correct choice. We should still check option (4) to confirm.

Analyze option (4): $6(x - 1) = 2(3x + 5)$

Distribute the $6$ on the left side and the $2$ on the right side of the equation.
$6x - 6 = 6x + 10$

Subtract $6x$ from both sides of the equation.
$6x - 6 - 6x = 6x + 10 - 6x$

Simplify both sides of the equation.
$-6 = 10$

Since $-6 \neq 10$ , this equation has no solution. Therefore, option (4) is not the correct choice.

After analyzing all options, we have determined that option (3) is the only equation with exactly one solution.
The equation with exactly one solution is: $5x + 2 = 3x + 14$