91Ó°ÊÓ

Linear equations are a type of equation where the highest power of the variable is one. They form straight lines when graphed.
In simpler terms, a linear equation is any equation that can be written in the form of \[Ax + By = C\].
Here, \(A\), \(B\), and \(C\) are constants.
The variables,\(x\) and\(y\), represent coordinates on a two-dimensional graph.
A key thing to remember about linear equations is they produce straight line graphs, and their solutions can be any point along this line.
These equations help us find relationships between quantities, like distance over time or cost per item.
When working with these, always aim to get to the slope-intercept form to better understand the slope and intercept.

The slope-intercept form of a linear equation is \( y = mx + b \).
In this formula:\

• \(y\) represents the dependent variable
• \(m\) stands for the slope of the line
• \(x\) is the independent variable
• \(b\) is the y-intercept

The slope (\(m\)) specifies how steep the line is.
The y-intercept (\(b\)) tells us where the line crosses the y-axis.
When given an equation, it's highly helpful to convert it into the slope-intercept form.
It makes it easier to graph and understand the relationship between variables.
If \(m\) is positive, the line slopes upward.
If \(m\) is negative, the line slopes downward.
In our exercise, we converted the original equation to the slope-intercept form \(y = 0x + \frac{9}{4}\).
This shows that the slope is \(0\), meaning it's a horizontal line.

To find the slope of a given linear equation, solving for y is crucial.
This involves isolating \(y\) on one side of the equation. In our exercise, the original equation:\[4y - 3x = 9 - 3x\]was simplified by adding \(3x\) to both sides, giving us:\[4y = 9\]Next, we divided both sides by 4:
\(y = \frac{9}{4}\)
By doing this, we isolated \(y\) to show the final form \(y = 0x + \frac{9}{4}\). This standard y = mx + b format allows us to easily identify the slope (\(m\)) and y-intercept (\(b\)).
Always remember that solving for y typically involves simplifying the equation by applying operations like addition, subtraction, multiplication, and division systematically.
These steps help in transforming the equation into a more useful form for interpretation.