Problem 56 - Elementary Algebra

Question

For Problems 55-70, solve each equation for the indicated variable. (Objective 4) $$ 5 x+2 y=12 \quad \text { for } x $$

Step-by-Step Solution

Answer

\(x = \frac{12 - 2y}{5}\)

Step 1: Isolate the Term with x

Begin by isolating the term with the variable you are solving for, which is x in this case. We have the equation: \(5x + 2y = 12\). Subtract \(2y\) from both sides to get: \(5x = 12 - 2y\).

Step 2: Solve for x

To solve for \(x\), divide each term by the coefficient of \(x\), which is 5. So divide the entire equation by 5: \(x = \frac{12 - 2y}{5}\). This results in \(x = \frac{12}{5} - \frac{2y}{5}\).

Key Concepts

Equation Solving

Equation solving is a fundamental aspect of algebra where the goal is to find the value of a variable that makes the equation true. The process involves manipulating the equation to isolate the desired variable, resulting in a clear solution. Solving equations can sometimes seem daunting, but understanding the basic steps can demystify the process.
To solve an equation, it’s important to perform the same operation on both sides of the equation. This ensures that the equation remains balanced. Imagine each side of the equation as weights on a scale; they must always be equal.

• Begin by simplifying each side of the equation, if necessary. This may involve combining like terms or applying the distributive property.
• Get all terms containing the variable to one side and all constant terms to the other.
• Use inverse operations to solve for the variable. Inverse operations are operations that undo each other, like addition and subtraction or multiplication and division.

Remember, practice makes perfect. With each equation you solve, the process becomes more intuitive.

Algebra Basics

Algebra is like a language that describes mathematical ideas. At its core, algebra involves working with symbols, typically letters called variables, to represent numbers in equations. Understanding the basics of algebra is crucial for solving equations and setting the foundation for more advanced mathematics.
Variables represent unknown values that we are interested in finding. Algebra allows us to express real-world situations and solve problems using these variables.

• An equation is a mathematical statement where two expressions are equal, connected by an equal sign.
• The goal is to manipulate the equation to uncover the value of the unknown variable.
• Key operations include addition, subtraction, multiplication, and division, which help in simplifying and solving equations.

Understanding these algebra basics is essential as they often build the groundwork for problem-solving in mathematics and many real-life applications.

Variable Isolation

Variable isolation is the process of rearranging an equation to have the variable you're interested in on one side and everything else on the other side. This process is key to solving equations because it directly leads to finding the value of the variable.
In our original exercise, the task was to solve for the variable \(x\) in the equation \(5x + 2y = 12\). To do this effectively, follow these steps:

• First, identify the term containing the variable you want to solve for and isolate it. This can mean moving other terms to the opposite side of the equation using inverse operations.
• In our equation, we moved \(2y\) to the right side by subtracting it from both sides.
• Next, eliminate any coefficients attached to the variable. Here, you would divide all terms by 5, the coefficient of \(x\), to isolate \(x\).

The result is an expression for \(x\) in terms of \(y\), showing the importance of variable isolation for solving equations and understanding relationships between variables.