Problem 42 - Algebra A Combined Function

Question

Solve each equation for \(y .\) $$ x=4 y+7 $$

Step-by-Step Solution

Answer

The solution is \\(y = \frac{x-7}{4}\\).

Step 1: Isolate the variable term

First, isolate the term containing the variable you want to solve for, which in this case is \(4y\). Start by subtracting \(7\) from both sides of the equation: \(x - 7 = 4y\).

Step 2: Solve for the variable

To solve for \(y\), divide both sides of the equation by \(4\): \(\frac{x-7}{4} = y\).

Key Concepts

Isolating Variables

When solving equations, one of the most vital steps is isolating the variable. This means getting the variable you want to solve for, by itself, on one side of the equation. Isolation is essential because it simplifies the problem and allows you to clearly see what the variable equals.
In the example equation, \( x = 4y + 7 \), our goal is to isolate the term \( 4y \) to eventually solve for \( y \). You start by removing anything that’s added or subtracted from \( y \).
Here are some key points to remember when isolating variables:

• Use addition or subtraction to eliminate constants from the side with the variable. For instance, subtract 7 from both sides to get \( x - 7 = 4y \).
• Keep the equation balanced. What you do to one side, you must do to the other.
• After isolating the term with \( y \), proceed to solve for \( y \).

This systematic approach of isolating helps in methodically breaking down problems.

Linear Equations

Linear equations are a type of algebraic equation. They make a straight line when graphed, hence the name "linear." These equations have no exponents higher than one and typically involve one or more variables like \( x \) and \( y \).
In the equation \( x = 4y + 7 \), you witness a classic structure of a linear equation. This equation can be rearranged to the form \( y = mx + b \), commonly known as the slope-intercept form. Here, \( m \) represents the slope and \( b \) the y-intercept.
Linear equations are fundamental in algebra because they form the basis of more complex topics. Understanding their structure and behavior is essential:

• The variable's highest power is 1 (there are no squares like \( y^2 \), cubes, etc.).
• Operations on a linear equation are straightforward involving addition, subtraction, multiplication, or division.
• Graphing gives insight into relationships between variables, making interpretation easier.

Hence, mastering linear equations opens doors to more advanced mathematical concepts.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying algebraic expressions to solve equations. This skill is fundamental in algebra because it allows you to express equations in different forms, ultimately leading to the solution.
Following the example equation \( x = 4y + 7 \), the manipulation starts by subtracting 7 to clear it from the side with \( y \).
Next, you divide by 4 to solve for \( y \) completely:

• Subtract constants or terms from both sides to move them. This simplification is crucial when working on one term at a time.
• Divide or multiply both sides of the equation as needed to isolate the variable. Using division with \( 4 \) results in \( y = \frac{x - 7}{4} \).
• Always check your work by plugging the variable back into the original equation to ensure it satisfies the equation as intended.

The art of algebraic manipulation makes it possible to handle more complicated equations systematically, turning complex mathematical statements into simpler, more workable forms.