Question

How do you write $ - 12x - 2y \geqslant 42 $ in slope-intercept form?

Answer 1

Hint: In order to determine the slope and intercept to the above inequality, we have to rewrite the inequality such that the LHS of the inequality contain only the $ y $ term by applying some operations. Add $ + 12x $ on both sides, then divide the both sides with the number $ - 2 $ but remember to reverse the inequality symbol to maintain the balance. Compare the result with the slope-intercept form $ y = mx + c $ , m is the slope and c is the y-intercept.

Complete step-by-step answer:
We are given a linear inequality in two variables $ x\,and\,y $ i.e. $ - 12x - 2y \geqslant 42 $
The slope-intercept of a linear equation is $ y = mx + c $ , where m is the slope of the line and c is the y-intercept of the graph.
Since, we are given a linear inequality, we have to rewrite the inequality in such a way that it can be comparable with the slope intercept form.
And do so we have to make the LHS of the inequality contain only the $ y $ term by applying some operations.
 $ - 12x - 2y \geqslant 42 $
Adding both sides of the inequality with $ + 12x $ . Remember that addition of any positive term on both sides of inequality does not affect the symbol of inequality.
 $
   - 12x - 2y + 12x \geqslant 42 + 12x \\
   - 2y \geqslant 12x + 42 \;
  $
Now dividing both sides of the inequality with the coefficient of variable $ y $ i.e. \[ - 2\].But wait , recall the rule of inequality that when a negative term is divided on both sides it disbalances the inequality. So, to balance it again we have to reverse the symbol of inequality as from $ \geqslant \,to\, \leqslant $ .
 $
  \dfrac{{ - 2y}}{{ - 2}} \leqslant \dfrac{{12x + 42}}{{ - 2}} \\
  y \leqslant - \dfrac{{12x}}{2} - \dfrac{{42}}{2} \\
  y \leqslant - 6x - 21 \\
  $
Comparing the above with slope-intercept form $ y = mx + c $
So $
  m = - 6 \\
  c = - 21 \;
  $
Therefore the required slope-intercept form is $ y \leqslant - 6x - 21 $ with slope $ m = - 6 $ and intercept as $ c = - 21 $
So, the correct answer is “ $ c = - 21 $ ”.

Note: 1. Graph of the inequality given in this question will have solid line as it contains the Slack inequality $ \left( { \geqslant , \leqslant } \right) $
2.Slope of line perpendicular to the line having slope $ m $ is equal to $ - \dfrac{1}{m} $ .
3.when a negative term is added on both sides then also the inequality symbol gets reversed.