91Ó°ÊÓ

Linear equations are fundamental in algebra. They are equations where all terms are either constants or products of constants and a variable, generally expressed in the form:

• \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants.

In simple terms, a linear equation will graph as a straight line. Each different linear equation corresponds to a line with different properties, such as slope and intercepts.
Take a look at the equation provided in the exercise: \(2x + 3y = 6\). This is a classic example of a linear equation. It is composed of two variables \(x\) and \(y\) and some constants. Understanding linear equations is key as they lay the groundwork for studying more complex algebraic concepts. Learning to manipulate these equations helps in describing various relationships between quantities in both science and daily life.

The slope-intercept form of a linear equation is a very useful way to express lines and identify properties easily. It is presented as:

• \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept.

This format is tremendously helpful because it directly reveals the slope of the line and its intersection point with the y-axis.
For the exercise, we transformed the equation \(2x + 3y = 6\) into slope-intercept form. After rearranging, we obtained \(y = -\frac{2}{3}x + 2\).
Here, \(-\frac{2}{3}\) is the slope of the line. This tells us that for every unit increase in \(x\), \(y\) decreases by \(\frac{2}{3}\). The intercept \(2\) implies the line crosses the y-axis at this point.

Algebraic manipulation is the process used to solve equations or rearrange them into a more helpful form. This involves several operations, including addition, subtraction, multiplication, division, and factoring. Such skills allow us to move from the general form of a linear equation to the slope-intercept form.
Let's review how we changed \(2x + 3y = 6\) to \(y = -\frac{2}{3}x + 2\):

• We subtracted \(2x\) from both sides, ensuring we maintain equality.
• Next, we divided every term by \(3\) to isolate \(y\), resulting in the slope-intercept form.

Being able to manipulate equations is crucial, not just to solve problems, but to understand the relationships between different elements of an equation. This skill grows with practice and lays a foundation for tackling more advanced mathematical problems.