Problem 14 - College Algebra with Corequisite Support

Question

For each of the following exercises, solve the equation for y in terms of \(x\). $$ x-2 y=7 $$

Step-by-Step Solution

Answer

The solution for \( y \) in terms of \( x \) is \( y = \frac{x - 7}{2} \).

Step 1: Isolate the Variable Term

To solve the equation for \( y \), we first need to make the terms involving \( y \) the subject of the equation. Start by subtracting \( x \) from both sides of the equation:\[ x - 2y = 7 \rightarrow -2y = 7 - x \]

Step 2: Solve for y

Now, we need to solve for \( y \). Since \( y \) is multiplied by \(-2\), divide both sides of the equation by \(-2\) to isolate \( y \):\[ y = \frac{7 - x}{-2} \]

Step 3: Simplify the Expression

Simplify the expression from Step 2:\[ y = \frac{7 - x}{-2} \] can be rewritten as:\[ y = \frac{x - 7}{2} \] by swapping the order of terms in the numerator and factoring out the negative sign.

Key Concepts

Isolating Variables

Isolating a variable is the process of manipulating an equation so that a specific variable, usually the one you want to solve for, stands alone on one side of the equation. In our given equation, \( x - 2y = 7 \), we are tasked with solving for \( y \) in terms of \( x \). The first step is to isolate the term containing \( y \). By subtracting \( x \) from both sides of the equation, we can move the \( x \) term to the opposite side, leaving only the \( -2y \) term separate:

• Original: \( x - 2y = 7 \)
• Subtract \( x \) from both sides: \( -2y = 7 - x \)

This step is crucial because it begins to focus the equation on \( y \), allowing us to target the term with further operations. Isolating variables simplifies the process of solving the equation.

Algebraic Manipulation

Algebraic manipulation involves a variety of techniques used to rearrange and simplify equations. Once a variable term is isolated, like \(-2y\) in this instance, the next step is to solve for the variable itself. This requires additional algebraic manipulation. In our example, \( y \) is being multiplied by \(-2\). To solve for \( y \), you'll need to divide both sides of the equation by \(-2\), making \( y \) alone on one side:

• Equation before manipulation: \( -2y = 7 - x \)
• Divide every term by \(-2\): \( y = \frac{7 - x}{-2} \)

This operation helps to further focus the equation on \( y \), removing other coefficients and simplifying the expression.

Equation Transformation

Equation transformation is a technique used to format an equation into a more useful form. After isolating \( y \) and using algebraic manipulation to solve for it, simplifying the expression can make it more interpretable. In our case, we have reached \( y = \frac{7 - x}{-2} \). By recognizing the need for simplification, you can transform the equation:

• Initial form: \( y = \frac{7 - x}{-2} \)
• Switch the terms and factor out \(-1\): \( y = \frac{x - 7}{2} \)

This transformation step changes the equation to a more straightforward form. It helps in perceiving the relationship between \( x \) and \( y \) more clearly, making the equation practical for further applications or evaluations.