Answer: The required answers are
(a) the slope of the given line is ![-\dfrac{3}{7}.](https://tex.z-dn.net/?f=-%5Cdfrac%7B3%7D%7B7%7D.)
(b) y-intercept exists and is equal to 6.
(c) the slope-intercept form of the line is ![y=-\dfrac{3}{7}x+6.](https://tex.z-dn.net/?f=y%3D-%5Cdfrac%7B3%7D%7B7%7Dx%2B6.)
Step-by-step explanation: We are given the following linear equation in two variables :
![3x+7y=42~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)](https://tex.z-dn.net/?f=3x%2B7y%3D42~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~%28i%29)
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
where m is the slope and c is the y-intercept of the line.
From equation (i), we have
![3x+7y=42\\\\\Rightarrow 7y=-3x+42\\\\\Rightarrow y=\dfrac{-3x+42}{7}\\\\\\\Rightarrow y=-\dfrac{3}{7}x+6.](https://tex.z-dn.net/?f=3x%2B7y%3D42%5C%5C%5C%5C%5CRightarrow%207y%3D-3x%2B42%5C%5C%5C%5C%5CRightarrow%20y%3D%5Cdfrac%7B-3x%2B42%7D%7B7%7D%5C%5C%5C%5C%5C%5C%5CRightarrow%20y%3D-%5Cdfrac%7B3%7D%7B7%7Dx%2B6.)
Comparing with the slope-intercept form, we get
![\textup{slope, m}=-\dfrac{3}{7},\\\\\\\textup{y-intercept, c}=6.](https://tex.z-dn.net/?f=%5Ctextup%7Bslope%2C%20m%7D%3D-%5Cdfrac%7B3%7D%7B7%7D%2C%5C%5C%5C%5C%5C%5C%5Ctextup%7By-intercept%2C%20c%7D%3D6.)
Thus,
(a) the slope of the given line is ![-\dfrac{3}{7}.](https://tex.z-dn.net/?f=-%5Cdfrac%7B3%7D%7B7%7D.)
(b) y-intercept exists and is equal to 6.
(c) the slope-intercept form of the line is ![y=-\dfrac{3}{7}x+6.](https://tex.z-dn.net/?f=y%3D-%5Cdfrac%7B3%7D%7B7%7Dx%2B6.)