To determine which form of a linear equation [tex]\( y = mx + b \)[/tex] represents, let's analyze the components of the equation:
- [tex]\( y \)[/tex]: This is the dependent variable, representing the value of the function at a given point.
- [tex]\( x \)[/tex]: This is the independent variable.
- [tex]\( m \)[/tex]: This is the slope of the line, indicating how steep the line is. It represents the rate of change of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex].
- [tex]\( b \)[/tex]: This is the y-intercept of the line, which is the point where the line crosses the y-axis when [tex]\( x = 0 \)[/tex].
This form of the equation is used to directly express the slope and the y-intercept of the line, which are key characteristics of the line's graph.
The multiple-choice options given are:
A. Parallel form
B. Slope-intercept form
C. Standard form
D. Point-slope form
Now, let's briefly describe each of these forms to identify which one fits [tex]\( y = mx + b \)[/tex]:
A. Parallel form: There is no common mathematical term known as the "parallel form" for linear equations.
B. Slope-intercept form: This form explicitly shows the slope and the y-intercept of the line. It is written as [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
C. Standard form: This form is generally written as [tex]\( Ax + By = C \)[/tex], where [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] are constants. It does not directly show the slope and the y-intercept, but it can be converted to slope-intercept form.
D. Point-slope form: This form is written as [tex]\( y - y_1 = m (x - x_1) \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( (x_1, y_1) \)[/tex] is a specific point on the line. This form is useful when you have a point and the slope but not the y-intercept directly.
Given this information, the form [tex]\( y = mx + b \)[/tex] matches the description of the Slope-intercept form.
Therefore, the correct answer is:
B. Slope-intercept form
