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Chapter 9: Problem 38

Solve each equation for \(y\). $$x=y-5$$

Short Answer

Expert verified
The solution is \(y = x + 5\).

Step by step solution

01

Isolate the Variable

The first step in solving this equation is to isolate the variable you want to solve for, which is \(y\). We start with the equation \(x = y - 5\). In order to isolate \(y\), we need to get rid of \(-5\) on the right side.
02

Add 5 to Both Sides

To eliminate \(-5\) from the right side, add 5 to both sides of the equation. This will give you: \ x + 5 = y - 5 + 5 \ Simplifying the right side, you get: \ x + 5 = y \ This isolates \(y\) on one side of the equation.

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Key Concepts

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

Isolating Variables
Isolating variables is a critical step in solving algebraic equations. The goal is to get the variable of interest by itself on one side of the equation. This helps us understand the relationship between the variables involved. Let's look at this process in more detail:
  • Identify the variable to isolate. In our example, it's the variable \( y \) in the equation \( x = y - 5 \).

  • Decide what operations are needed to isolate the variable. Here, the operation required is to "undo" subtraction.

  • Perform the inverse operation. For subtraction, the inverse is addition. Thus, you add 5 to both sides to cancel out the \(-5\).
Adding 5 to both sides of the equation ensures that the variable \( y \) stands alone, simplifying the expression to \( x + 5 = y \). This clear isolation of the variable allows us to easily solve or analyze the equation.
Linear Equations
Linear equations are basic yet vital components of algebra. They form straight lines when graphed and include variables raised to the first power. Understanding linear equations helps in various applications, from business forecasting to physics.
  • The general form of a linear equation is \( ax + b = 0 \) or \( y = mx + c \), where \( m \) represents the slope and \( c \) is the y-intercept.

  • In the example \( x = y - 5 \), we have a straightforward equation representing a line where \( y \) is expressed in terms of \( x \).

  • These equations are predictable and can be solved to find unknown values defining specific points along a line.
When dealing with linear equations, every operation you carry out is aimed at maintaining the equality while restructuring the expression for simplicity or practical interpretation.
Algebraic Manipulation
Algebraic manipulation is the process of rearranging equations to solve for a specific variable. This involves using mathematical operations strategically to transform equations while keeping them equivalent. Let's explore this further:
  • Start by looking at the given equation and decide which operation can simplify it or help isolate the variable.

  • Use arithmetic operations such as addition, subtraction, multiplication, or division to achieve the desired form. In our equation \( x = y - 5 \), we used addition.

  • Remember to perform the chosen operation on both sides of the equation to maintain balance and equivalence.
Through algebraic manipulation, equations can be reshaped to emphasize particular relationships or to find solutions efficiently and clearly. Mastery of this skill is essential for solving complex mathematical problems.

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Most popular questions from this chapter

Solve. See Example 4. $$ \log _{2} 8=x $$

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