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Chapter 2: Problem 368

Solve each equation with decimal coefficients. $$0.05(q-5)+0.25 q=3.05$$

Short Answer

Expert verified
q = 11

Step by step solution

01

Distribute the Coefficient

Distribute the coefficient of 0.05 to both terms inside the parentheses: \( 0.05(q-5) = 0.05q - 0.25 \)
02

Combine Like Terms

Rewrite the equation with the distributed term: \( 0.05q - 0.25 + 0.25q = 3.05 \) Then combine the like terms: \( 0.05q + 0.25q - 0.25 = 3.05 \) \( 0.30q - 0.25 = 3.05 \)
03

Isolate the Variable Term

Add 0.25 to both sides of the equation to isolate the term with the variable: \( 0.30q = 3.05 + 0.25 \) \( 0.30q = 3.30 \)
04

Solve for q

Divide both sides of the equation by 0.30 to solve for \( q \): \( q = \frac{3.30}{0.30} \) \( q = 11 \)

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Key Concepts

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

Distributive Property
To solve equations with decimals, it's crucial to understand the distributive property.
The distributive property is a way to simplify expressions and make equations easier to solve.
It states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products.
For example, in the equation, \[ 0.05(q-5) + 0.25q = 3.05 \], we distribute the 0.05 across each term inside the parentheses.
This results in: \[ 0.05(q-5) = 0.05q - 0.25 \]. This step helps to eliminate the parentheses and simplify the equation.
Combining Like Terms
Combining like terms is another essential skill when solving equations.
Like terms are terms that have the same variable raised to the same power.
In our example: \[ 0.05q - 0.25 + 0.25q = 3.05 \], the like terms are \(0.05q\) and \(0.25q\).
Adding them together, we get \(0.30q - 0.25 = 3.05\). Combining like terms simplifies the equation making it easier to isolate the variable in the next steps.
Isolating the Variable
Isolating the variable means getting the variable term on one side of the equation by itself.
To isolate \( q \) in our equation \( 0.30q - 0.25 = 3.05 \), we need to move the constant term (-0.25) to the other side of the equation.
We do this by adding 0.25 to both sides: \[ 0.30q - 0.25 + 0.25 = 3.05 + 0.25 \]. Simplifying, we get: \[ 0.30q = 3.30 \]. With the variable term isolated, we're ready for the final step鈥攕olving for the variable.
Dividing Decimals
The last step in the process involves dividing decimals.
To solve \( 0.30q = 3.30 \), we need to divide both sides of the equation by the coefficient of \( q \), which is 0.30.
So we have: \[ q = \frac{3.30}{0.30} \].
To simplify, remember that dividing by a decimal is the same as multiplying both the numerator and the denominator by the same power of 10 to make the denominator a whole number. In this case, \( q = 11 \). This final step gives us the solution to our original equation.

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