Answer: The required answers are
(a) the slope of the given line is [tex]-\dfrac{3}{7}.[/tex]
(b) y-intercept exists and is equal to 6.
(c) the slope-intercept form of the line is [tex]y=-\dfrac{3}{7}x+6.[/tex]
Step-by-step explanation: We are given the following linear equation in two variables :
[tex]3x+7y=42~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
We are to :
(a) determine the slope,
(b) determine the y-intercept, if exists
and
(c) express equation in slope-intercept form.
We know that
The SLOPE_INTERCEPT form of the equation of a straight line is given by
[tex]y=mx+c,[/tex] where m is the slope and c is the y-intercept of the line.
From equation (i), we have
[tex]3x+7y=42\\\\\Rightarrow 7y=-3x+42\\\\\Rightarrow y=\dfrac{-3x+42}{7}\\\\\\\Rightarrow y=-\dfrac{3}{7}x+6.[/tex]
Comparing with the slope-intercept form, we get
[tex]\textup{slope, m}=-\dfrac{3}{7},\\\\\\\textup{y-intercept, c}=6.[/tex]
Thus,
(a) the slope of the given line is [tex]-\dfrac{3}{7}.[/tex]
(b) y-intercept exists and is equal to 6.
(c) the slope-intercept form of the line is [tex]y=-\dfrac{3}{7}x+6.[/tex]
