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Chapter 2: Problem 40

Solve each formula for the indicated letter. $$ x+10 y=2, \text { for } y $$

Short Answer

Expert verified
y = \frac{2 - x}{10}

Step by step solution

01

Isolate the Term with y

Start by isolating the term that contains the variable you want to solve for. In this case, it's the term with \(y\). To do this, subtract \(x\) from both sides of the equation:\[x + 10y = 2\] becomes \[10y = 2 - x\]
02

Solve for y

Now, solve for \(y\) by dividing both sides of the equation by 10 to isolate \(y\):\[10y = 2 - x\] Dividing both sides by 10 gives:\[y = \frac{2 - x}{10}\]

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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 core concept in algebra. It means rearranging an equation to get a single variable by itself on one side of the equation. This process helps to solve for that variable. For instance, if we are given an equation with multiple terms, such as in our example \(x + 10y = 2\), we want to solve for \(y\). The first step is to isolate the term with \(y\).

This involves moving any other terms to the opposite side of the equation. Here, we subtract \(x\) from both sides, resulting in \[10y = 2 - x\]. By isolating \(y\), we simplify the process of solving for it.
Linear Equations
Linear equations are equations of the first order. They involve variables which are not raised to any power other than one. A linear equation in one variable looks like \(ax + b = 0\), and in two variables, like \(ax + by = c\). In our example, the equation \(x + 10y = 2\) is a linear equation involving two variables, \(x\) and \(y\).

These equations produce straight lines when graphed. The key to working with linear equations is understanding that they represent a balance. Whatever operation you perform on one side must be done to the other side. This concept is crucial in solving linear equations.
Algebraic Manipulation
Algebraic manipulation refers to the techniques we use to rearrange and simplify equations. By applying operations such as addition, subtraction, multiplication, and division, we transform equations into simpler forms. This makes it easier to isolate variables and eventually solve the equations.

In our example, after isolating \(10y\) by subtracting \(x\) from both sides, we further manipulate the equation by dividing both sides by 10. This algebraic manipulation gives us \[ y = \frac{2 - x}{10}\]

These steps show how algebraic manipulations assist us in turning complex equations into manageable forms, helping us solve for the desired variable.

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