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Chapter 2: Problem 14

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

Short Answer

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

Step by step solution

01

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:\[ x - 2y = 7 \rightarrow -2y = 7 - x \]
02

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 \( y \):\[ y = \frac{7 - x}{-2} \]
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

For the following exercises, solve the inequality. Write your final answer in interval notation. $$ 3 x+2 \geq 7 x-1 $$

For the following exercises, solve the inequality. Write your final answer in interval notation. $$ 4 x-7 \leq 9 $$

For the following exercises, graph the function. Observe the points of intersection and shade the \(x\) -axis representing the solution set to the inequality. Show your graph and write your final answer in interval notation. $$ |x-1|>2 $$

For the following exercises, solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions. $$ \sqrt{t+1}+9=7 $$

For the following exercises, write the set in interval notation. $$ \\{x \mid x<4\\} $$
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