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Chapter 2: Problem 57

Solve each equation, and check your solution. $$ 5.2 q-4.6-7.1 q=-0.9 q-4.6 $$

Short Answer

Expert verified
The solution is q = 0.

Step by step solution

01

- Simplify both sides of the equation

First, simplify both sides of the equation by combining like terms. On the left side, combine the terms involving q: \[ 5.2q - 7.1q - 4.6 = -0.9q - 4.6 \] This simplifies to: \[ -1.9q - 4.6 = -0.9q - 4.6 \]
02

- Get all q terms on one side

Next, add 1.9q to both sides of the equation to get all terms involving q on one side: \[ -1.9q + 1.9q - 4.6 = -0.9q + 1.9q - 4.6 \] This simplifies to: \[ -4.6 = 1q - 4.6 \]
03

- Isolate the variable

Add 4.6 to both sides to isolate the variable q: \[ -4.6 + 4.6 = 1q - 4.6 + 4.6 \] This simplifies to: \[ 0 = q \]
04

- Check your solution

Substitute q = 0 back into the original equation to verify: \[ 5.2 \times 0 - 4.6 - 7.1 \times 0 = -0.9 \times 0 - 4.6 \] This simplifies to: \[ -4.6 = -4.6 \] Both sides are equal, so the solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combining Like Terms
Combining like terms is a crucial step in solving linear equations. It involves adding or subtracting terms that have the same variables. For instance, in the equation given, both 5.2q and -7.1q are like terms because they both have the variable q.
To simplify, you need to combine these terms:
\[5.2q - 7.1q = -1.9q \] After this step, our equation becomes \[-1.9q - 4.6 = -0.9q - 4.6\].
This makes the equation easier to handle by reducing the number of terms. Always remember: like terms must have the same variable and exponent. Only then can they be combined.
Isolating the Variable
Once we have combined like terms, the next step is to isolate the variable q. This means getting q on one side of the equation by itself. In our equation \[-1.9q - 4.6 = -0.9q - 4.6\], we need to eliminate the q term on one side.
We do this by adding 1.9q to both sides:
\[-1.9q + 1.9q - 4.6 = -0.9q + 1.9q - 4.6\]
This simplifies to:
\[-4.6 = 1q - 4.6\] Now, we add 4.6 to both sides to isolate q:
\[-4.6 + 4.6 = 1q - 4.6 + 4.6\]
This simplifies to:
\1q = 0\.
Lastly, we write q = 0. Isolating the variable ensures that we know the exact value of q that satisfies the equation.
Checking Solutions
After you find the value of the variable, checking your solution is crucial to verify its accuracy. We substitute the solution back into the original equation and see if both sides are equal.
Substitute q = 0 back into the original equation:
\[5.2q - 4.6 - 7.1q = - 0.9q - 4.6\]
Replacing q with 0:
\[5.2(0) - 4.6 - 7.1(0) = - 0.9(0) - 4.6\]
This simplifies to:
\[-4.6 = - 4.6\]
As both sides are equal, the solution q = 0 is verified to be correct. Always remember to check your solutions as it helps confirm the correctness of your work and catch any potential errors.

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