Problem 60 - Elementary Algebra

Question

For Problems 55-70, solve each equation for the indicated variable. (Objective 4) $$ -x+14 y=17 \quad \text { for } x $$

Step-by-Step Solution

Answer

The solution for \(x\) is \(x = 14y - 17\).

Step 1: Identify the Given Equation

The given equation is \(-x + 14y = 17\). We need to solve for \(x\). This means we'll rearrange the equation to express \(x\) in terms of \(y\).

Step 2: Isolate the Term with x

Add \(x\) to both sides of the equation to eliminate the negative sign in front of \(x\). This gives us: \(14y = x + 17\).

Step 3: Solve for x

Subtract 17 from both sides to isolate \(x\) on one side of the equation. This results in: \(x = 14y - 17\).

Key Concepts

Solving for a Variable

When dealing with algebraic equations, one common task is to solve for a specific variable. This involves finding the expression for that variable in terms of the other variables present in the equation. The goal is to isolate the variable of interest, often using basic arithmetic operations like addition, subtraction, multiplication, or division. In our example, we need to solve for \(x\) in the equation \(-x + 14y = 17\). By rearranging the equation, \(x\) is expressed solely in terms of \(y\). Solving for a variable becomes handy when you need to understand how it changes with respect to other variables.

Understanding Linear Equations

Linear equations are foundational in algebra and present the simplest form for equations: they graph as straight lines in the coordinate plane. The standard form of a linear equation in two variables is \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants. In our specific scenario, the equation \(-x + 14y = 17\) is a linear equation with two variables, \(x\) and \(y\). These equations are called 'linear' because they produce linear graphs, simplifying relationships between variables and making them easy to solve through straightforward algebraic manipulations.

Technique of Equation Rearrangement

Equation rearrangement is a technique used to isolate a desired variable on one side of the equation. This is achieved through a sequence of algebraic operations that might involve moving terms from one side of the equation to the other. Rearranging equations allows you to intuitively understand how changes in one variable affect others.

• First, identify what term needs to be isolated - in our exercise, it's \(x\).
• Next, use inverse operations to move other terms to the opposite side. For example, add \(x\) to both sides to eliminate the negative sign: \(14y = x + 17\).
• Lastly, perform one final operation to achieve expression isolation: subtract 17 from both sides resulting in \(x = 14y - 17\).

By mastering equation rearrangement, you gain a powerful mathematical tool that aids in solving not only linear equations but myriad other algebraic problems.