91Ó°ÊÓ

The slope-intercept form of a linear equation is a very common way to express linear equations. It is written as:

• y = mx + b

Here, \( y \) represents the dependent variable, \( x \) is the independent variable, \( m \) is the slope of the line, and \( b \) is the y-intercept. The form is handy because it allows you to quickly identify the slope and the y-intercept just by looking at the equation. This is useful when you need to draw the graph of the line or understand how it behaves.
For example, in the equation \( y = \frac{1}{3}x - \frac{7}{3} \), the slope \( m \) is \( \frac{1}{3} \), which means the line rises one unit for every three units it runs to the right. The y-intercept \( b \) is \( -\frac{7}{3} \), which means the line crosses the y-axis at the point \( (0, -\frac{7}{3}) \).

Solving linear equations usually involves isolating the variable you're solving for on one side of the equation. Let's break it down with the given example:
The original equation is \( x = 3y + 7 \).

1. To isolate \( y \), start by subtracting 7 from both sides:

• \( x - 7 = 3y \)

2. Next, divide both sides by 3 to get \( y \) by itself:

• \( y = \frac{x - 7}{3} \)

This isolates \( y \) and puts the equation into a form that can be easily converted to slope-intercept form.
Finally, rewrite it as \( y = \frac{1}{3}x - \frac{7}{3} \) so it looks like \( y = mx + b \).This makes it simple to identify the slope and intercept.

Identifying the slope and y-intercept is straightforward when you are dealing with the slope-intercept form of the linear equation \( y = mx + b \). Here’s how you do it for our example:

• The slope \( m \) tells you how steep the line is and the direction it tilts. If \( m \) is positive, the line slopes upwards as you move from left to right. If \( m \) is negative, the line slopes downwards.
• In our equation \( y = \frac{1}{3}x - \frac{7}{3} \), the slope \( m \) is \( \frac{1}{3} \). This means for every 3 units you move to the right (the x-direction), the line rises 1 unit (the y-direction).
• The y-intercept \( b \) is the value of \( y \) when \( x \) is 0. In our case, \( b \) is \( -\frac{7}{3} \). This means the line crosses the y-axis at \( -\frac{7}{3} \).

These simple tricks help you draw the graph and understand the linear equation's behavior without complex calculations.