Problem 19 - Elementary Algebra

Question

For exercises 11-30, (a) solve. (b) check. $$ \frac{15}{4 z}+\frac{2}{3}=\frac{1}{24} $$

Step-by-Step Solution

Answer

z = -6.

Step 1 - Identify the Equation

We are given the equation: \[ \frac{15}{4z} + \frac{2}{3} = \frac{1}{24} \]

Step 2 - Find Common Denominator

To eliminate fractions, find a common denominator for the terms. The denominators are 4z, 3, and 24. The common denominator is 24z.

Step 3 - Multiply the Entire Equation

Multiply each term by the common denominator, 24z: \[ 24z \times \frac{15}{4z} + 24z \times \frac{2}{3} = 24z \times \frac{1}{24} \]

Step 4 - Simplify Each Term

Simplify the equation by canceling out the denominators: \[ 6 \times 15 + 8z \times 2 = z \]

Step 5 - Combine Like Terms

Combine and simplify the terms: \[ 90 + 16z = z \]

Step 6 - Solve for z

Isolate the variable z by subtracting z from both sides: \[ 90 + 15z = 0 \]Then solve for z: \[ 15z = -90 \]\[ z = -6 \]

Step 7 - Check the Solution

Substitute z back into the original equation to verify: Original equation: \[ \frac{15}{4(-6)} + \frac{2}{3} = \frac{1}{24} \]Calculate each term: \[ \frac{15}{-24} + \frac{2}{3} = -\frac{5}{8} + \frac{16}{24} = \frac{-5}{8} + \frac{2}{3} \]Find common denominator for -5/8 and 2/3: \[ = -\frac{20}{24} + \frac{16}{24} = \frac{-4}{24} = \frac{-1}{6} \]Since neither side equals each other, there is an error in solving. Reassess steps for calculation mistakes.

Key Concepts

Finding Common Denominator

In order to add or subtract fractions, we need to have a common denominator. This helps in eliminating the fractions, making the equation easier to solve. Let's take a look at the rational equation: \[ \frac{15}{4z} + \frac{2}{3} = \frac{1}{24} \]The denominators here are 4z, 3, and 24. To find the common denominator, we must identify the least common multiple (LCM) of these denominators.

• 4z: A variable with a coefficient of 4.
• 3: A prime number.
• 24: A multiple of 4 (4 * 6), and it includes factors of 3 (3 * 8).

Looking at these, we notice that 24 includes factors of both 3 and 4. By assigning z to the LCM, we get 24z as the common denominator. This allows us to multiply each term by 24z to eliminate the fractions.

Simplifying Rational Expressions

Now, let's simplify the equation by multiplying every term by the common denominator, 24z. This step is key to eliminating the fractions from our equation. \[ 24z \times \frac{15}{4z} + 24z \times \frac{2}{3} = 24z \times \frac{1}{24} \] Simplifying each term involves canceling out the denominators:

• For the first term, 24z \times \( \frac{15}{4z} \) = 6 \times 15 = 90.
• For the second term, 24z \times \( \frac{2}{3} \) = 8z \times 2 = 16z
• For the third term, 24z \times \( \frac{1}{24} \) = 1z = z

This simplifies our equation to: \[ 90 + 16z = z \]Note that simplifying these expressions correctly is crucial because any mistake can lead to an incorrect solution.

Solving for a Variable

Now that we've simplified the equation to 90 + 16z = z, the next step is to isolate the variable z to find its value. First, let's get all instances of z on one side of the equation. Subtract z from both sides: \[ 90 + 16z - z = 0 \] which simplifies to: \[ 90 + 15z = 0 \] Next, we need to isolate z. Subtract 90 from both sides: \[ 15z = -90 \] Finally, divide both sides by 15 to solve for z: \[ z = \frac{-90}{15} \] \[ z = -6 \] So, the value of z is -6. Always double-check that the steps are correctly followed to ensure the solution is accurate.

Checking Solutions in Algebra

After solving for z, it is crucial to check your solution to ensure that no errors were made during the simplification and solving process. To do this, substitute z back into the original equation: \[ \frac{15}{4(-6)} + \frac{2}{3} = \frac{1}{24} \]Calculate each term:

• \( \frac{15}{-24} = -\frac{5}{8} \)
• \( \frac{2}{3} = \frac{16}{24} \)

Thus, the equation becomes: \[ -\frac{5}{8} + \frac{16}{24} = \frac{-5}{8} + \frac{2}{3} \]Finding the common denominator for -\( \frac{5}{8} \) and \( \frac{2}{3}\): \( = -\frac{20}{24} + \frac{16}{24} = \frac{-4}{24} = \frac{-1}{6} \)However, if the sides don’t match, there might be an error in the solution process. Reassess each step carefully to identify any mistakes.