Problem 81 - Elementary Algebra

Question

Solve for the indicated variable. $$ \text { Solve for } x: z=2 x+y s $$

Step-by-Step Solution

Answer

\( x = \frac{z - ys}{2} \)

Step 1: Understand the Equation

We are given the equation \( z = 2x + ys \). We need to solve this equation for \( x \), meaning we aim to express \( x \) in terms of the other variables \( z, y, \) and \( s \).

Step 2: Isolate Terms Involving x

To solve for \( x \), we first need to isolate the terms involving \( x \) on one side of the equation. We can do this by subtracting \( ys \) from both sides of the equation:\[ z - ys = 2x \]

Step 3: Solve for x

Now that we have the equation \( z - ys = 2x \), we will solve for \( x \) by dividing both sides by 2:\[ x = \frac{z - ys}{2} \]

Step 4: Final Solution

We have successfully isolated \( x \) and expressed it in terms of the other variables. The solution for \( x \) is:\[ x = \frac{z - ys}{2} \]

Key Concepts

Isolating Variables

When solving equations, a common goal is to find one specific variable by itself, also known as isolating the variable. In our exercise, we are tasked with solving for \( x \) in the equation: \[ z = 2x + ys \]To isolate \( x \), it's important to rearrange the terms so that \( x \) stands alone on one side of the equation. We begin this process by making sure that all terms involving \( x \) are placed on one side. In this instance, we subtract \( ys \) from both sides:\[ z - ys = 2x \]This manipulation successfully isolates the terms with \( x \), bringing us closer to expressing \( x \) independently. This step is crucial because it simplifies the equation, making it easier to focus directly on the variable of interest. If you ever need to isolate a different variable, the same principle applies: gather all expressions involving the desired variable on one side of the equation.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging equations to reveal the desired information. Once we isolate the terms with \( x \), the equation becomes simpler:\[ z - ys = 2x \]From here, algebraic manipulation involves straightforward operations like addition, subtraction, multiplication, and division. In this scenario, our aim is to solve for \( x \), which means we need \( x \) by itself.To do this, we leverage division:

• Notice that \( 2x \) means \( x \) is multiplied by 2.
• To undo this multiplication, divide both sides of the equation by 2.

This gives us:\[ x = \frac{z - ys}{2} \]These manipulations are a vital part of solving equations, allowing us to isolate variables and express them in a clear, concise form. This process may involve more complex steps for more intricate equations, but the core principles remain consistent.

Expressing Variables in Terms of Others

Expressing a variable in terms of others is a common step in solving equations, especially when dealing with formulas or multi-variable equations. Here, the final outcome of our manipulations is that we have expressed \( x \) as a function of three other variables, \( z \), \( y \), and \( s \):\[ x = \frac{z - ys}{2} \]This means \( x \) is defined by these three variables and showcases a relationship between them. If you know the values of \( z \), \( y \), and \( s \), you can calculate the value of \( x \) directly using this expression.Expressing variables in terms of others is particularly useful:

• In rearranging formulas to find unknown quantities.
• In programming, where functions often depend on multiple variables.
• In sciences, where equations need to be redefined to understand variable dependencies.

This approach helps in understanding the dynamics between different variables and can simplify complex problems into manageable pieces.