To determine whether the statement "The equation of a line in slope-intercept form is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the [tex]\( x \)[/tex] intercept" is true or false, we need to examine the components of the slope-intercept form of a linear equation.
The slope-intercept form of a line is given by:
[tex]\[ y = mx + b \][/tex]
Here:
- [tex]\( y \)[/tex] represents the dependent variable (usually the vertical axis).
- [tex]\( x \)[/tex] represents the independent variable (usually the horizontal axis).
- [tex]\( m \)[/tex] is the slope of the line, which describes its steepness and direction.
- [tex]\( b \)[/tex] is the y-intercept, the point where the line crosses the y-axis.
Key points to note:
1. The slope [tex]\( m \)[/tex] denotes the rate of change of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex]. It is the ratio of the vertical change to the horizontal change between any two points on the line.
2. The y-intercept [tex]\( b \)[/tex] is the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex].
The equation [tex]\( y = mx + b \)[/tex] does not specify anything about the [tex]\( x \)[/tex]-intercept directly. The x-intercept, where the line crosses the x-axis, is the value of [tex]\( x \)[/tex] for which [tex]\( y = 0 \)[/tex]. To find the x-intercept from the slope-intercept form, set [tex]\( y = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = mx + b \][/tex]
[tex]\[ x = -\frac{b}{m} \][/tex]
Therefore, [tex]\( m \)[/tex] is the slope of the line, not the [tex]\( x \)[/tex]-intercept.
Given this analysis, the statement "The equation of a line in slope-intercept form is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the [tex]\( x \)[/tex] intercept" is incorrect.
Thus, the correct answer is:
B. False
