Problem 31 - College Algebra and Calculus: An Applied Approach

Question

Solve the equation and check your solution. (Some equations have no solution.) $$ 0.25 x+0.75(10-x)=3 $$

Step-by-Step Solution

Answer

The solution to the equation \(0.25 x + 0.75(10 - x) = 3\) is \(x = 9\).

Step 1: Simplify the Equation

To start, simplify the equation by distributing the \(0.75\) to the \(10-x\) in the equation. This results in \(0.25x + 7.5 - 0.75x = 3\). Then, combine like terms on the left side to create a simpler equation.

Step 2: Solve for 'x'

The equation now is \(-0.5x + 7.5 = 3\). To isolate 'x', you first need to move '7.5' to the right side of the equation by subtracting '7.5' from both sides: \(-0.5x = 3 - 7.5\) which simplifies to \(-0.5x = -4.5\). To solve for 'x', divide both sides of the equation by '-0.5', so you get \(x = -4.5 / -0.5\).

Step 3: Check the Solution

After calculating, you find that \(x = 9\). Check this solution by substituting \(9\) for \(x\) in the original equation and confirm if it balances. Substituting gives \(0.25(9) + 0.75(10 - 9) = 3\), which simplifies to \(2.25 + 0.75 = 3\), verifying that \(x = 9\) is indeed the correct solution.

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols such as plus, minus, multiply, and divide. For instance, in the equation from our exercise, we encounter algebraic expressions like 0.25x and 0.75(10-x). Understanding these expressions is crucial when solving linear equations, as they represent quantities and their relationships.

To manipulate these expressions effectively, it is important to comprehend concepts like distribution, where we multiply a constant through a bracket, and combining like terms, which involves grouping and simplifying terms that have the same variable to their highest power. In our example, we distributed the 0.75 across the 10 - x to get 7.5 - 0.75x, turning the original expression into a simpler form that's easier to work with.

Equation Simplification

The process of equation simplification involves rewriting an equation to make it easier to solve. This usually begins with expanding brackets, if any, and then combining like terms. For equations that involve decimals, such as the one we're working with (0.25x + 7.5 - 0.75x = 3), it's helpful to line up the decimal points carefully when performing arithmetic operations.

When we simplify by combining like terms, we reduced the equation to -0.5x + 7.5 = 3, stripping away some complexity and making the 'x' term easier to isolate. Simplifying the equation is a critical step towards finding the solution, as it breaks down the problem into more manageable parts.

Checking Solutions

After solving a linear equation, it's essential to check the solution to ensure it's correct. This involves substituting the solution back into the original equation and verifying whether the left and right sides equal. In our exercise, when the value x = 9 is plugged into the equation, it simplifies to 2.25 + 0.75 = 3, the left side matching the right side.

Checking not only verifies the accuracy of the solution but also reinforces understanding of how algebraic expressions work within equations. It's a confirmation that all the steps in the solving process – from simplification to solving – were performed correctly.