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To graph a linear equation, we first need to rewrite it in slope-intercept form. This form is given by the equation: \[ y = mx + b \] Here, \( m \) represents the slope of the line and \( b \) is the y-intercept.

• The slope \( m \) tells us the steepness and direction of the line.
• The y-intercept \( b \) specifies where the line crosses the y-axis.

For example, given the equation \( 6x - 5y = 18 \), we must first manipulate it to fit the slope-intercept form. By doing some algebra:
Subtract \( 6x \) from both sides: \[ -5y = -6x + 18 \] Then, divide each term by \( -5 \): \[ y = \frac{6}{5}x - \frac{18}{5} \] Now, the equation is in the form \( y = mx + b \), where \( m = \frac{6}{5} \) and \( b = -\frac{18}{5} \). This form makes it easier to identify the slope and y-intercept for graphing.

The slope \( m \) of a line reflects how steep the line is and the direction it goes. In the slope-intercept formula \( y = mx + b \), the slope is given directly by the coefficient of \( x \).

• A positive slope indicates the line rises as it moves from left to right.
• A negative slope shows the line falls as it moves from left to right.

For instance, in the equation \( y = \frac{6}{5}x - \frac{18}{5} \), the slope \( m = \frac{6}{5} \) means the line rises by 6 units for every 5 units it moves to the right. This ratio helps us properly graph the line by determining how quickly it rises or descends.

To graph a line, we need to plot points based on the equation. The y-intercept \( b \) is a starting point where the line crosses the y-axis, and from there, we use the slope to find more points.

• Start by plotting the y-intercept. For the equation \( y = \frac{6}{5}x - \frac{18}{5} \), the y-intercept is \( b = -\frac{18}{5} \). So, plot the point \( (0, -\frac{18}{5}) \) where the line crosses the y-axis.

Now, the slope tells us how to find another point. With a slope of \( \frac{6}{5} \), for every 5 units right (positive x direction), we move 6 units up (positive y direction). From the y-intercept of \( (0, -\frac{18}{5}) \), moving to the right by 5 units leads us to \( x = 5 \), and then moving up 6 units calculates the y-coordinate: \( y = -\frac{18}{5} + 6 = -\frac{18}{5} + \frac{30}{5} = \frac{12}{5} \).
Plot the point \( (5, \frac{12}{5}) \). Now we have two necessary points to draw the line.

After plotting at least two points, we can now draw the line representing the equation.

• The points \( (0, -\frac{18}{5}) \) and \( (5, \frac{12}{5}) \) are already identified.

Use a ruler to make sure the line is straight, extending it through the plotted points.
Check the alignment by ensuring it adheres to the slope; the line should rise by 6 units for every 5 units it moves right. This straight line is the graphical representation of the equation \( 6x - 5y = 18 \).
By following these steps, you will be able to graph any linear equation, making complex concepts understandable and easier to manage!