Problem 45 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(43-50,\) solve each equation for \(x .\) $$y=(a-b) x+5$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = \frac{y-5}{a-b}\), with the condition that \(a\) should not be equal to \(b\), as it would make the denominator equal to zero, and the equation would not be defined.

Step 1. Rearrange the Equation

To begin, we start by moving the 5 from the right-hand side to the left-hand side. We do this by subtracting 5 from both sides, effectively isolating the \(x\) term. The equation then becomes \(y-5 = (a-b) x\).

Step 2. Solve for x

To isolate \(x\), we can divide the entire equation by \(a-b\), assuming a is not equal to b as it would make the denominator zero. so, \(x = \frac{y-5}{a-b}\).

Step 3: Final Answer

Hence, the solution for \(x\) in terms of \(y\), \(a\), and \(b\) is \(x = \frac{y-5}{a-b}\). Note: This only holds as long as \(a\) is not equal to \(b\). If \(a=b\), then the equation does not hold true as the denominator becomes zero, leading to an undefined result.

Key Concepts

Algebraic Manipulations

Solving linear equations efficiently requires a firm grasp of algebraic manipulations. These are the arithmetic operations you perform on an equation to transform it into a simpler form. The goal is to rearrange the terms and make the equation more manageable without changing its solutions. For example, when we subtract 5 from both sides of the equation y = (a-b)x + 5, we are performing an algebraic manipulation.

Key processes in algebraic manipulations include:

• Adding or subtracting the same value from both sides to eliminate or introduce terms.
• Multiplying or dividing both sides by the same non-zero value to simplify coefficients or isolate the variable.
• Distributing a value across a set of parentheses or factoring a common element out of a set of terms.

These skills are essential since they allow us to transform complex expressions into simpler ones that are easier to solve.

Isolate Variable

The heart of solving a linear equation is to isolate the variable you're solving for. This means manipulating the equation so that the variable appears by itself on one side of the equation. In our exercise, we aim to isolate x. After subtracting 5 from both sides, the next step is to divide through by the coefficient (a-b).

Isolating the variable is a multi-step process that might involve:

• Clearing fractions or decimals
• Combining like terms
• Using inverse operations to cancel out addition, subtraction, multiplication, or division

When isolating x as in our example, we make sure that the operation we perform does not lead to division by zero, thereby ensuring the equation remains valid.

Equation Solving Steps

Solving an equation can be thought of as a journey where each step brings you closer to the destination—the solution. The equation solving steps provide a structured approach. Let's review the steps exemplified in solving the given exercise:

Identify and Simplify

First, identify the variable to solve for and simplify the equation if needed. This includes clearing parentheses and combining like terms.

Isolate the Variable

Next, use algebraic manipulations to get the variable on one side of the equation. This might require multiple operations. In our example, subtracting and then dividing were necessary.

Perform the Operation

Finally, perform the necessary arithmetic operation. Ensuring that the operations are mathematically valid, such as avoiding division by zero, is crucial.

Each of these steps ensures clarity and reduces the chance of errors, leading to the correct solution: x = (y-5)/(a-b), provided that a ≠ b to avoid an undefined denominator.