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Chapter 2: Problem 116

Solve each equation requiring simplification. $$\frac{5}{12} q+\frac{1}{2} q=25-3$$

Short Answer

Expert verified
The solution is \( q = 24 \).

Step by step solution

01

Combine like terms on the left side

First, we need to combine the fractions \( \frac{5}{12} q \) and \( \frac{1}{2} q \) on the left side of the equation. To do this, we find a common denominator. The least common denominator of 12 and 2 is 12. Rewrite \( \frac{1}{2} \) as \( \frac{6}{12} \). Therefore, the equation becomes: \[ \frac{5}{12}q + \frac{6}{12}q = 25 - 3 \]
02

Simplify the equation

Combine the fractions on the left side by adding their numerators: \[ \frac{5+6}{12}q = 25 - 3 \] This simplifies to: \[ \frac{11}{12}q = 22 \]
03

Solve for q

To isolate \( q \), multiply both sides of the equation by the reciprocal of \( \frac{11}{12} \), which is \( \frac{12}{11} \): \[ q = 22 \times \frac{12}{11} \] Simplifying this gives us: \[ q = 2 \times 12 = 24 \]

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Key Concepts

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

Combining Like Terms
When solving equations, combining like terms can simplify the equation and make it easier to solve. Like terms are terms with the same variable raised to the same power.
In our exercise, we have two terms with the variable \(q\): \( \frac{5}{12}q \) and \( \frac{1}{2}q \).
To combine these, they must have the same denominator.
Here's how to do it step-by-step:
  • Find the least common denominator of the fractions.
  • Rewrite each term with this common denominator.
  • Add the coefficients of the like terms together.
After combining like terms on the left side, we turn \( \frac{5}{12}q + \frac{1}{2}q \) into \( \frac{11}{12}q \).
This makes the equation simpler to handle.
Least Common Denominator
The least common denominator (LCD) is the smallest number that can be a common denominator for two or more fractions.
In solving equations with fractions, finding the LCD makes it easier to combine fractions.
Let's find the LCD for our exercise step-by-step:
  • List the denominators: 12 and 2.
  • Determine the smallest multiple that both numbers share.
  • For 12 and 2, the smallest multiple is 12.
Now we rewrite \( \frac{1}{2} \) as \( \frac{6}{12} \),
This allows us to easily add it to \( \frac{5}{12} \) since they share the same denominator. This concept is vital when simplifying equations with fractions.
Isolating Variables
Isolating the variable means getting it by itself on one side of the equation.
This makes it possible to solve for the variable.
Here's how to do it in our exercise:
  • We began with: \( \frac{11}{12}q = 22 \).
  • To isolate \( q \), we multiply both sides by the reciprocal of \( \frac{11}{12} \), which is \( \frac{12}{11} \).
This step effectively eliminates the fraction that is attached to the variable.
So, the equation becomes: \( q = 22 \times \frac{12}{11} \).
Simplifying the right side, we get \( q = 24 \).
By mastering the technique of isolating variables, you can solve any equation efficiently.
Always ensure to perform the same operation on both sides of the equation to maintain balance.

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