91Ó°ÊÓ

The slope-intercept form of an equation is one of the most common ways to represent a linear equation. It is written as:
\[ y = mx + b \]
Where:

• \( y \): the dependent variable (usually y)
• \( m \): the slope of the line
• \( x \): the independent variable (usually x)
• \( b \): the y-intercept — the point where the line crosses the y-axis

To find the slope-intercept form of a line, you need to know the slope and the y-intercept. For example, in the exercise given, converting the standard form equation 3x - y = 7 to slope-intercept form involves isolating y on one side:
\[ 3x - y = 7 \]
Subtract 3x from both sides:
\[ -y = -3x + 7 \]
Next, multiply both sides by -1 to solve for y:
\[ y = 3x - 7 \]
Now, the equation is in slope-intercept form, and you can see that the slope \( m \) is 3 and the y-intercept \( b \) is -7.

The point-slope form is a handy format for writing the equation of a line when you know one point on the line and the slope. It is given by:
\[ y - y_1 = m(x - x_1) \]
Where:

• \( (x_1, y_1) \): a point on the line
• \( m \): the slope of the line

This form is particularly useful when you are working with parallel lines since parallel lines have the same slope. In the exercise, to find the equation of the line parallel to 3x - y = 7 and passing through the point (1, 3), you use the point-slope form:
You have:

• Point \( (x_1, y_1) = (1, 3) \)
• Slope \( m = 3 \)

Substitute these values into the point-slope form:
\[ y - 3 = 3(x - 1) \]
This gives you an equation of the line. Next, you simplify it to your desired form, either slope-intercept or standard form.

The standard form of a linear equation is another way to represent a line. It is written as:
\[ Ax + By = C \]
Where:

• \( A, B, \) and \( C \) are integers

Sometimes, converting between different forms of linear equations is necessary. In the exercise, after using point-slope and converting to slope-intercept form, let's now convert the equation y = 3x to standard form.
Start from the slope-intercept form:
\[ y = 3x \]
Subtract 3x from both sides to move it to the left side:
\[ -3x + y = 0 \]
If desired, you can multiply through by -1 to have the x term positive:
\[ 3x - y = 0 \]
Now, this is the standard form of the line equation. Remember, the standard form doesn't change the line's properties, just how it's represented.