Problem 24 - Prealgebra and Introductory Algebra

Question

For Exercises 21 to \(32,\) solve for \(y\). $$5 x-y=7$$

Step-by-Step Solution

Answer

The solution to the equation for \(y\) is \(y = 5x - 7\).

Step 1: Isolate \(y\) on one side of the equation

Start by adding \(y\) to both sides of the equation to cancel out the \( - y\) on the left side. This transforms the equation from \(5x - y = 7\) to \(5x = y + 7\).

Step 2: Solve for \(y\)

Now, subtract \(7\) from both sides of the equation to isolate \(y\). By doing this, the equation transforms from \(5x = y + 7\) to \(y = 5x - 7\).

Key Concepts

Isolate Variable

Understanding how to isolate a variable is a fundamental skill in algebra, particularly when solving linear equations. When you isolate a variable, you rearrange the equation so that the variable you're solving for is on one side of the equal sign and everything else is on the other side. This is usually achieved through a series of operations that are applied equally to both sides of the equation to maintain balance. For example, in the equation 5x - y = 7, the goal is to get y by itself. By adding the y term to both sides, we effectively move it to the other side of the equation, leaving 5x alone on the other side, thus performing the initial step to isolate y.

Here's why this method is so powerful: it simplifies the equation to a point where the solution for the variable can be directly read off, providing you with the relationship between the variable and the other terms in the equation. This technique not only applies to linear equations but also serves as a cornerstone for dealing with more complex algebraic expressions.

Algebraic Manipulation

Algebraic manipulation involves operations performed on an equation to change its form without affecting the solution set. The various operations include adding, subtracting, multiplying, dividing both sides of the equation by the same number, and applying functions to both sides. The golden rule of these operations is to always treat both sides equally to preserve the equation’s integrity. For instance, in our problem 5x = y + 7, subtracting 7 from both sides ensures that whatever value x may hold, the equality holds true. This step transforms the equation to its simplified form, y = 5x - 7.

Why is algebraic manipulation useful?

It allows for the equation to be reshaped into a simpler form that is more intuitive to understand and helps unveil the inherent relationships between variables. Additionally, it paves the way for solving equations that might otherwise seem too complicated to handle directly.

Equation Solving Steps

Solving linear equations typically follows a structured approach or a series of equation solving steps. The goal is to find the value(s) of the variable(s) that satisfy the equation. Initially, you will identify the variable to solve for and look to isolate this variable using algebraic manipulation. The steps for solving the equation 5x - y = 7 involve isolating y and resolving the equation to its simplest form.

Step 1: Add y to both sides of the equation, yielding 5x = y + 7.
Step 2: Subtract 7 from both sides, resulting in y = 5x - 7, where y is isolated.

Key Tips for Equation Solving Steps:

• Perform operations systematically to avoid confusion.
• Check your work by plugging the solution back into the original equation.
• Practice with a variety of equations to become familiar with different scenarios.

By consistently applying these steps, you'll become adept at simplifying and solving linear equations, enhancing both your confidence and competence in algebra.