Problem 80 - Elementary Algebra

Question

For exercises 79-82, (a) clear the fractions and solve. (b) check. $$ \frac{3}{2} u+\frac{3}{4}=\frac{9}{2} $$

Step-by-Step Solution

Answer

u = \( \frac{5}{2} \)

Step 1 - Clear the Fractions

To clear the fractions, find a common denominator for all the fractions in the equation. The denominators are 2 and 4. The least common multiple (LCM) is 4. Multiply every term in the equation by 4:distribute: \[ 4 \times \frac{3}{2}u + 4 \times \frac{3}{4} = 4 \times \frac{9}{2} \]This gives:\[ 2 \times 3u + 3 = 2 \times 9 \]which simplifies to:\[ 6u + 3 = 18 \]

Step 2 - Solve for u

Isolate the variable by subtracting 3 from both sides of the equation:\[ 6u = 18 - 3 \]This simplifies to:\[ 6u = 15 \]Next, divide both sides by 6:\[ u = \frac{15}{6} \]Simplify the fraction:\[ u = \frac{5}{2} \]

Step 3 - Check the Solution

Substitute the value of \( u \) back into the original equation to verify it:\[ \frac{3}{2} \times \frac{5}{2} + \frac{3}{4} = \frac{9}{2} \]First, simplify \( \frac{3}{2} \times \frac{5}{2} \):\[ \frac{15}{4} \]Then add \( \frac{3}{4} \):\[ \frac{15}{4} + \frac{3}{4} = \frac{18}{4} \]Simplify \( \frac{18}{4} \):\[ \frac{9}{2} \]Both sides of the original equation are equal, confirming that \( u = \frac{5}{2} \) is the correct solution.

Key Concepts

clearing fractions

Clearing fractions is the first crucial step when solving equations that involve fractions. Fractions can make equations difficult to handle directly, so the goal is to eliminate them. To do this, we find the least common multiple (LCM) of all the denominators involved. Once we have the LCM, we multiply every term in the equation by this LCM. For example, in our exercise, the denominators are 2 and 4, and the LCM is 4. By multiplying everything by 4, each fraction disappears, simplifying our equation.

least common multiple

The least common multiple (LCM) is key to clearing fractions. The LCM of two or more numbers is the smallest number that is evenly divisible by all of them. In our example, the denominators are 2 and 4. The LCM of 2 and 4 is 4 because it's the smallest number that both 2 and 4 divide into without leaving a remainder. By multiplying each term in the equation by the LCM, we can convert the fractional equation into a whole number equation, making it easier to solve.

isolating variables

Once the fractions are cleared, the next step is to isolate the variable. This means we want to get the variable by itself on one side of the equation. To do this, we perform operations that will cancel out the other terms. In our exercise, after clearing fractions, we have the equation 6u + 3 = 18. By subtracting 3 from both sides, we isolate the term with the variable: 6u = 15. After that, dividing both sides by 6 finishes the isolation process: u = 15/6.

simplifying fractions

Simplifying fractions is an important step in solving equations. It makes the numbers easier to work with and understand. After isolating the variable, sometimes the result is an improper fraction. For instance, u = 15/6. To simplify this, we find the greatest common divisor (GCD) of the numerator and denominator. Here, the GCD of 15 and 6 is 3. Dividing both the numerator and the denominator by 3, we get u = 5/2. This fraction is simpler and easier to interpret.

checking solutions

Always check your solutions to ensure accuracy. This means substituting the solution back into the original equation to see if it makes both sides equal. In our exercise, we found u = 5/2. We substitute it back: \(\frac{3}{2} * \frac{5}{2} + \frac{3}{4} = \frac{9}{2}\). Simplifying, \(\frac{15}{4} + \frac{3}{4} = \frac{18}{4}\), and \(\frac{18}{4} = \frac{9}{2}\). Both sides of the equation match, verifying that our solution u = 5/2 is correct. This step ensures that no mistakes were made during the solving process.