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Question Solve the equation 1.8q−1.5−4.6q=−3.8q−5.71.8 q - 1.5 - 4.6 q = -3.8 q - 5.7, and find the solution set.

Studdy Solution

STEP 1

Assumptions
1. We are given the equation 1.8q−1.5−4.6q=−3.8q−5.71.8q - 1.5 - 4.6q = -3.8q - 5.7.
2. We need to solve for the variable qq.
3. We will perform algebraic operations to isolate qq on one side of the equation.

STEP 2

Combine like terms on both sides of the equation involving qq.
1.8q−4.6q+3.8q=−5.7+1.51.8q - 4.6q + 3.8q = -5.7 + 1.5

STEP 3

Simplify the terms on both sides.
0.2q=−4.20.2q = -4.2

STEP 4

Divide both sides of the equation by the coefficient of qq to solve for qq.
q=−4.20.2q = \frac{-4.2}{0.2}

STEP 5

Calculate the value of qq.
q=−21q = -21

STEP 6

Check the solution by substituting q=−21q = -21 back into the original equation.
1.8(−21)−1.5−4.6(−21)=−3.8(−21)−5.71.8(-21) - 1.5 - 4.6(-21) = -3.8(-21) - 5.7

STEP 7

Perform the multiplication for each term.
−37.8−1.5+96.6=79.8−5.7-37.8 - 1.5 + 96.6 = 79.8 - 5.7

STEP 8

Combine the like terms on each side of the equation.
57.3=74.157.3 = 74.1

STEP 9

We notice that the check does not result in an equality. This indicates that there might have been a mistake in the calculation. We need to re-evaluate the steps to find and correct the error.

STEP 10

Re-check the multiplication of the coefficients and the variable qq.
1.8(−21)=−37.81.8(-21) = -37.8 −4.6(−21)=96.6-4.6(-21) = 96.6 −3.8(−21)=79.8-3.8(-21) = 79.8

STEP 11

Re-check the addition and subtraction of the constants.
−37.8−1.5=−39.3-37.8 - 1.5 = -39.3 96.6+79.8=176.496.6 + 79.8 = 176.4 −39.3+176.4=137.1-39.3 + 176.4 = 137.1 79.8−5.7=74.179.8 - 5.7 = 74.1

STEP 12

We realize that the mistake was in the addition and subtraction of the constants. Let's correct the calculation.
−37.8−1.5+96.6=−39.3+96.6-37.8 - 1.5 + 96.6 = -39.3 + 96.6

STEP 13

Perform the correct addition.
−39.3+96.6=57.3-39.3 + 96.6 = 57.3

STEP 14

Now, compare the corrected left side of the equation with the right side.
57.3=74.157.3 = 74.1

STEP 15

We notice that the check still does not result in an equality. This indicates that there might have been a mistake in the solution. Let's re-solve the equation from the beginning.

STEP 16

Re-write the original equation.
1.8q−1.5−4.6q=−3.8q−5.71.8q - 1.5 - 4.6q = -3.8q - 5.7

STEP 17

Combine like terms involving qq on both sides of the equation.
1.8q−4.6q+3.8q=−5.7+1.51.8q - 4.6q + 3.8q = -5.7 + 1.5

STEP 18

Simplify the terms on both sides.
0.2q=−4.20.2q = -4.2

STEP 19

Divide both sides of the equation by the coefficient of qq to solve for qq.
q=−4.20.2q = \frac{-4.2}{0.2}

STEP 20

Calculate the value of qq again.
q=−21q = -21

STEP 21

Check the solution again by substituting q=−21q = -21 back into the original equation.
1.8(−21)−1.5−4.6(−21)=−3.8(−21)−5.71.8(-21) - 1.5 - 4.6(-21) = -3.8(-21) - 5.7

STEP 22

Perform the multiplication for each term again.
−37.8−1.5+96.6=79.8−5.7-37.8 - 1.5 + 96.6 = 79.8 - 5.7

STEP 23

Combine the like terms on each side of the equation again.
57.3=74.157.3 = 74.1

STEP 24

We have made the same error again. Let's carefully re-evaluate the combination of like terms involving qq.

STEP 25

Re-combine the like terms involving qq on the left side of the equation, paying close attention to the signs.
1.8q−4.6q=−2.8q1.8q - 4.6q = -2.8q

STEP 26

Now, combine the like terms involving qq on both sides of the equation correctly.
−2.8q+3.8q=−5.7+1.5-2.8q + 3.8q = -5.7 + 1.5

STEP 27

Simplify the terms on both sides.
1.0q=−4.21.0q = -4.2

STEP 28

Now that we have the correct coefficient for qq, divide both sides of the equation by the coefficient of qq to solve for qq.
q=−4.21.0q = \frac{-4.2}{1.0}

STEP 29

Calculate the value of qq correctly.
q=−4.2q = -4.2

STEP 30

Check the solution one last time by substituting q=−4.2q = -4.2 back into the original equation.
1.8(−4.2)−1.5−4.6(−4.2)=−3.8(−4.2)−5.71.8(-4.2) - 1.5 - 4.6(-4.2) = -3.8(-4.2) - 5.7

STEP 31

Perform the multiplication for each term.
−7.56−1.5+19.32=15.96−5.7-7.56 - 1.5 + 19.32 = 15.96 - 5.7

STEP 32

Combine the like terms on each side of the equation.
10.26=10.2610.26 = 10.26

STEP 33

The check results in an equality, which means q=−4.2q = -4.2 is the correct solution.
The solution set is {−4.2}\{-4.2\}.

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