91Ó°ÊÓ

In solving linear equations, one of the key steps is isolating the variable you want to solve for. In the given exercise, our goal is to find the value of the variable \(y\). To do this, we first need to isolate the term that contains \(y\) on one side of the equation. This means we move all other terms to the opposite side.
Looking at the equation given: \[3x + 4y = 7\],
the term containing the \(y\) is \(4y\). To isolate \(4y\), we need to remove \(3x\) from the left side.
We achieve this by subtracting \(3x\) from both sides of the equation:
\[4y = 7 - 3x\]
The subtraction of \(3x\) from both sides leaves \(4y\) by itself on the left side.

A linear equation is any equation that can be written in the form \(ax + by = c\). This type of equation represents a straight line when plotted on a graph. In the exercise provided, \(3x + 4y = 7\) is an example of a linear equation. Here, \(a=3\) and \(b=4\) are coefficients, while \(c=7\) is the constant term.
Linear equations are beneficial because they are straightforward and predictable. The solutions to linear equations are the points where the line intersects the axes when graphed.
In solving the equation for \(y\), we change its form but its property as a linear equation remains. The final solution \(y = \frac{7 - 3x}{4}\) still represents a linear relationship between \(x\) and \(y\).

Algebraic manipulation involves using algebraic rules and operations to simplify or rearrange equations. These operations include addition, subtraction, multiplication, division, and applying the distributive property. In solving the given linear equation for \(y\), we use several key algebraic manipulations.
1. **Subtraction:** We start by subtracting \(3x\) from both sides to isolate the term with \(y\):
\[4y = 7 - 3x\]
2. **Division:** To solve for \(y\), we divide every term by 4:
\[y = \frac{7 - 3x}{4}\]
These steps are all about making the equation simpler and getting \(y\) by itself. Mastering these basic rules of algebraic manipulation is essential for solving more complex equations in algebra and beyond.