Problem 79 - Introductory and Intermediate Algebra for College Students 4th

Question

$$\text { Solve: } \frac{x}{2}+7=13-\frac{x}{4} . \text { (Section 2.3, Example 4) }$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 8\)

Step 1: Bring x terms on one side

To facilitate the process, we need to gather all terms involving x on one side and constants on the other side. This involves adding \(\frac{x}{4}\) to both sides and subtracting 7 from both sides. Doing this we get \(\frac{x}{2} + \frac{x}{4} = 13 - 7\)

Step 2: Convert fraction to simplest form

To simplify the equation further, make sure both sides of the equation have the same denominator. Here, the least common multiple of 2 and 4 is 4. Therefore, convert the fractions on left side to have the denominator of 4. We get \(\frac{2x}{4} + \frac{x}{4} = 6\)

Step 3: Simplify and Solve for x

Now simplify the left side of the equation, we get \(\frac{3x}{4} = 6\). Multiplying 4 on both sides and then dividing by 3 we get \(x = 8\)

Key Concepts

Fraction Addition

Fraction addition might sound scary, but it's easier than you think! The key to adding fractions is ensuring they share the same denominator. This is called finding a common denominator. Let's see how it applies in the equation \(\frac{x}{2} + \frac{x}{4} = 6\) from the exercise.

• First, identify the denominators in the fractions. Here, we have 2 and 4.
• The smallest number that both 2 and 4 divide into evenly is 4. This is the least common multiple (LCM), and you'll use it as the common denominator.
• Convert \(\frac{x}{2}\) into a fraction with a denominator of 4. To do this, multiply both numerator and denominator by 2 to get \(\frac{2x}{4}\).
• Once both fractions have the same denominator, add them together: \(\frac{2x}{4} + \frac{x}{4} = \frac{3x}{4}\).

Adding fractions becomes simple once you line up the denominators! This strategy works for complex calculations, too.

Combining Like Terms

Combining like terms is critical in simplifying algebraic equations. It's like finding match-ups among terms to make the equation neat and tidy.Think of like terms as terms within an equation that have the same variable raised to the same power. For example, in the equation \(\frac{2x}{4} + \frac{x}{4} = 6\), both terms \(\frac{2x}{4}\) and \(\frac{x}{4}\) carry the variable "x" to the same power, making them combinable.

• Add the coefficients (numbers in front of variables) of like terms together. Here, we add 2 and 1 (coefficient of \(\frac{x}{4}\) when the denominators are the same) to get \(3x\).
• Your equation should now look like \(\frac{3x}{4} = 6\) after combining.

By combining like terms correctly, you reduce the complexity of the equation and make solving it way simpler.

Solving Linear Equations

Solving linear equations involves finding the value of the unknown variable that makes the equation true. It entails getting \(x\) on its own side, free from all other numbers or operations.Using the example from our problem: \(\frac{3x}{4} = 6\)

• To isolate \(x\), begin by eliminating fractions. Multiply both sides by 4 (the denominator): \(3x = 24\).


• Next, divide both sides by the coefficient of \(x\), which is 3 in this instance: \(x = \frac{24}{3}\).


• After simplifying, you find that \(x = 8\).

Through these steps, you see the beauty of unraveling equations down to its simplest form. Linear equations are like puzzles: arrange the pieces logically to reveal the solution. Don't forget to check your answer by substituting it back into the original equation, verifying its accuracy!