Problem 8 - Algebra 2 and Trigonometry

Question

In \(3-20,\) solve each equation and check. $$ x-0.05 x=19 $$

Step-by-Step Solution

Answer

The solution is \(x = 20\).

Step 1: Simplify the Equation

Start by simplifying the left side of the equation by factoring out the common term. Notice that both terms on the left side of the equation have the variable \(x\). Write it as \( x(1 - 0.05) = 19 \).

Step 2: Combine Like Terms

Calculate \(1-0.05 = 0.95\). So, the equation becomes \(0.95x = 19\). This equation now represents a simple multiplication problem.

Step 3: Solve for x

To isolate \(x\), divide both sides of the equation by 0.95. This gives \(x = \frac{19}{0.95}\).

Step 4: Calculate the Division

Perform the division to find the value of \(x\): \(x = 20\).

Step 5: Check the Solution

Substitute \(x = 20\) back into the original equation to verify its correctness: \(20 - 0.05 \times 20 = 19\). Calculate the left-hand side: \(0.05 \times 20 = 1\), so \(20 - 1 = 19\). This confirms the solution is correct.

Key Concepts

Factoring

Factoring is an essential skill when solving algebraic equations, and it helps simplify them. In this context, factoring involves identifying a common factor in multiple terms. For instance, in the equation \(x - 0.05x = 19\), both terms on the left side, \(x\) and \(0.05x\), share the variable \(x\) as a factor. Factoring it out makes the expression easier to manage. To factor it out, you would write the expression as \(x(1 - 0.05)\). Essentially, you're acknowledging that each term involves \(x\), and makes it possible to reduce the expression using distributive properties. Factoring is a powerful tool because it often simplifies expressions to a single, straightforward operation. By simplifying the original equation, you make the rest of the solution much more accessible.

Simplification

Simplification is the process of making an algebraic expression easier to work with. After factoring, the next logical step is to simplify the expression fully. Let's look at \(x(1 - 0.05) = 19\). First, perform the subtraction inside the parentheses: calculate \(1 - 0.05\) to get \(0.95\). This reduces your equation to \(0.95x = 19\), which is much simpler and easier to handle. Simplification helps transform complex equations into more manageable forms. It involves combining like terms, simplifying fractions, and reducing unnecessary components to streamline the equation. This makes it easier to isolate variables and find solutions.

Multiplication and Division

Once you have simplified the equation to \(0.95x = 19\), it's time to solve for \(x\) using multiplication and division. This equation is straightforward because it is in the form of a simple multiplication. To isolate \(x\), divide both sides of the equation by \(0.95\). This step involves division, \(x = \frac{19}{0.95}\). Perform the division to find \(x\). In this scenario, dividing gives \(x = 20\). These steps illustrate how multiplication and division are integral to solving algebra equations. They allow you to move terms and simplify expressions to solve for unknowns.

Solution Verification

Solution verification is a necessary step to ensure accuracy in your algebraic solutions. Once you've found \(x = 20\), substitute it back into the original equation to check its validity.Start by substituting \(x = 20\) into the equation: \(20 - 0.05 \times 20 = 19\). Compute \(0.05 \times 20\) to get \(1\), then calculate the left-hand side: \(20 - 1 = 19\). The equation holds true, confirming that your solution is correct. Verification not only ensures accuracy but also builds confidence in your arithmetic and algebraic manipulations. It is a good mathematical habit that helps avoid and correct potential mistakes.