Problem 92 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether the given number is a solution of the equation. $$\frac{5}{3} x=30 ; 18$$

Step-by-Step Solution

Answer

Yes, 18 is a solution to the given equation.

Step 1: Identify the given number and the equation

The provided number is 18 and the given equation is \( \frac{5}{3}x = 30 \).

Step 2: Substitute the given number in the equation

Place the value of 18 in place of x in the equation. This yields \( \frac{5}{3} * 18 \).

Step 3: Calculate the value

Perform the multiplication to get \( \frac{5}{3} * 18 = 30 \).

Step 4: Check if the obtained value matches the right-hand side of the equation

Looking at the right-hand side of the equation, we see it equals to 30. Thus, the left-hand side when calculated also equals to 30.

Step 5: Conclusion

As the left-hand side and right-hand side of the equation are both equal to 30, it can be determined that the number 18 is indeed a solution to the given equation.

Key Concepts

Equation Solving

Equation solving is a fundamental concept in algebra where we find the value of the variable that makes the equation true. When an equation is set, it balances a mathematical statement on either side of the equal sign. To solve an equation, our goal is to find the value of the unknown variable that satisfies this balance.

Understanding how to solve equations is crucial because it allows us to find unknown values systematically.

• Identify the equation: Observe the given mathematical statement.
• Perform operations: Use mathematical operations to isolate the variable on one side of the equation.
• Verify solution: Substitute the value back into the original equation to check if both sides match.

This process not only helps us in finding solutions but also enhances logical and analytical thinking skills used across many fields of study.

Substitution Method

The substitution method is a handy technique used in equation solving, typically when dealing with multiple equations. In this approach, you substitute one part of the equation with a specific value to simplify the process.

For the current problem, the substitution method was employed to check if the number 18 is a solution.

• Substitute: Replace the variable in the equation with the given number, in this case, 18.
• Calculation: Perform the arithmetic to see if the equation holds true.

This method simplifies finding solutions by breaking down complex equations into manageable pieces. Particularly in linear equations, substitution is a straightforward way to test possible solutions and verify their correctness. Additionally, the method is vital when solving systems of equations, enabling us to express terms in relation to each other.

Linear Equations

Linear equations are equations where the highest power of the variable is one—meaning there are no squares, cubes, or higher powers involved. They appear in the form of Ax + B = C, where A, B, and C are constants. Linear equations are straightforward and consist of simple relationships between different quantities.

In our example, the given equation \( \frac{5}{3}x = 30 \) is a linear equation. Linear equations represent lines when plotted graphically and are foundational for more complex concepts in mathematics.

• Simplify: Simplifying linear equations often involves basic arithmetic operations.
• Balance: Maintain the balance of the equation by performing equivalent operations on both sides.

Understanding linear equations is crucial because they are widely used in various fields, including science, engineering, and economics, to model relationships between variables and predict outcomes.