Problem 40 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve equation and check your proposed solution. Begin your work by rewriting each equation without fractions. \(\frac{z}{5}-\frac{1}{2}=\frac{z}{6}\)

Step-by-Step Solution

Answer

The solution to the equation is \(z=15\).

Step 1: Remove fractions

First, we find the least common multiple of 5, 2, and 6, which is 30. Multiply every term in the equation by 30. The equation becomes: \(30*\frac{z}{5}-30*\frac{1}{2}=30*\frac{z}{6}\) This simplifies to: \(6z-15=5z\)

Step 2: Simplify and solve the equation

Next, organize the equation and solve for \(z\). Subtract \(5z\) from both sides of the equation: \(6z-5z-15=0\), which simplifies to \(z-15=0\). Then add 15 to both sides to solve for \(z\): \(z=15\)

Step 3: Check the solution

Lastly, we will substitute \(z\) with 15 in the original equation to check if the solution is correct: \(\frac{15}{5}-\frac{1}{2}=\frac{15}{6}\) \(3-0.5=2.5\). This solves to \(2.5=2.5\). Since both sides of the equation are equal, the solution \(z=15\) is correct.

Key Concepts

Algebra

Algebra involves using symbols, like letters, to represent numbers in equations. In the given exercise, we have an equation with the variable \(z\). Our goal is to find the value of \(z\) that makes the equation true.
By manipulating the equation using algebraic rules, we can isolate \(z\) on one side, allowing us to see what it's equal to. This process involves operations like addition, subtraction, multiplication, and division.
Remember, what you do to one side of the equation, you must do to the other to keep things balanced. The balance is crucial in algebra because it ensures that the equality remains true throughout all the steps of solving.

Fractions

Fractions are a way of representing parts of a whole. In the exercise, the fractions \(\frac{z}{5}\), \(\frac{1}{2}\), and \(\frac{z}{6}\) must be dealt with before proceeding with the solution.
Working with fractions can be a bit tricky, but they are just another way to represent numbers. When solving equations with fractions, it helps to eliminate them to simplify the equation.
You can do this by finding a common denominator or using the least common multiple (LCM) to clear the fractions, which brings us to the next topic.

Least Common Multiple

The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. To eliminate the fractions in our equation, we find the LCM of the denominators: 5, 2, and 6.
The LCM of 5, 2, and 6 is 30 because 30 is the smallest number divisible by all these denominators.

• Use 30 to multiply each term in the equation.
• This process is called "clearing the fractions" because it transforms the equation into one without fractional coefficients.

By converting the equation to whole numbers, it becomes much easier to solve.

Equation Checking

Checking your solution is a critical step in solving equations. Once we have a solution, like \(z = 15\), we need to verify it's correct.
Substitute \(z = 15\) back into the original equation: \(\frac{15}{5} - \frac{1}{2} = \frac{15}{6}\).

• Simplify each side of the equation to ensure they are equal.
• If both sides are equal, your solution is verified; otherwise, it indicates a mistake in your work.

Checking helps catch any errors and confirms the solution's accuracy, providing confidence in solving the equation correctly.