Problem 51 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve equation and check your proposed solution in. \(0.3 x-4=0.1(x+10)\)

Step-by-Step Solution

Answer

The solution to the equation \( 0.3 x-4=0.1(x+10) \) is \( x = 25 \).

Step 1: Distribute the variable on the right side

Distribute \( x \) on the right side of the equation: \( 0.3 x - 4 = 0.1x + 1 \).

Step 2: Rearrange equation to isolate \( x \)

Subtract \( 0.1x \) from both sides of the equation: \( 0.3x - 0.1x = 1 + 4 \). Simplifying the equation gives \( 0.2x = 5 \).

Step 3: Solve for \( x \)

To get the value of \( x \), divide by \( 0.2 \) on both sides: \( x = 5 / 0.2 \). Calculating gives \( x = 25 \).

Step 4: Validate solution

Substitute \( x = 25 \) back into the original equation to check the solution: \( 0.3(25) - 4 =? 0.1(25 + 10) \). Simplifying both sides gives \( 7.5 = 7.5 \), hence confirming the solution as valid one.

Key Concepts

Algebraic Expressions

Algebraic expressions are crucial in solving linear equations as they represent relationships using numbers and variables. Variables act like placeholders, typically representing unknown values that we aim to find. In the original exercise, the expression on the left side of the equation, \( 0.3x - 4 \), integrates both a coefficient (0.3) and a constant term (-4) along with the variable. The right side, \( 0.1(x + 10) \), includes a variable term with a coefficient, and an addition within the parentheses.

Understanding how to handle these expressions is essential. It allows us to manipulate the equations correctly to isolate the variable, which is the next important concept in this context. By applying distribution to \( 0.1(x+10) \), we expand it to \( 0.1x + 1 \), making it easier to rearrange the equation in subsequent steps.

Variable Isolation

Variable isolation is a technique we use to rearrange an equation so that the variable we need to solve for stands alone on one side of the equation. This process requires understanding and applying several algebraic operations, like addition, subtraction, multiplication, and division.

In the equation \( 0.3x - 4 = 0.1x + 1 \), the task is to get \( x \) by itself. By subtracting \( 0.1x \) from both sides, we combine like terms to simplify the equation to \( 0.2x = 5 \). This step-by-step rearrangement involves simplification and helps focus on the variable. Once \( 0.2x = 5 \) is simplified, dividing both sides by 0.2 completes the isolation, providing \( x = 25 \).

• Subtract or add terms to simplify equations.
• Use inverse operations to isolate the variable.

Mastering variable isolation enables solving complex algebraic equations step by step.

Equation Validation

Equation validation is the process of ensuring that the solution we have found actually satisfies the original equation. It's a crucial step to confirm accuracy in solving any algebraic problem.

Once we find \( x = 25 \), this validation involves substituting 25 back into the original equation: \( 0.3(25) - 4 \) should equal \( 0.1(25 + 10) \). Working through this computation confirms both sides equal 7.5, hence validating our solution.

Why validate? It ensures

• Accuracy: Confirming the solution is correct.
• Understanding: Prove that the operations and rearrangements executed were properly applied.

By practicing equation validation, we build confidence in our algebraic manipulation skills, which are essential for tackling more advanced problems.