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Question: How do you write \( - 12x - 2y \geqslant 42 \) in slope-intercept form?...

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

Explanation

Solution

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 yy term by applying some operations. Add +12x+ 12x on both sides, then divide the both sides with the number −2- 2 but remember to reverse the inequality symbol to maintain the balance. Compare the result with the slope-intercept form y=mx+cy = 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 yx\,and\,y i.e. −12x−2y⩾42- 12x - 2y \geqslant 42
The slope-intercept of a linear equation is y=mx+cy = 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 yy term by applying some operations.
−12x−2y⩾42- 12x - 2y \geqslant 42
Adding both sides of the inequality with +12x+ 12x . Remember that addition of any positive term on both sides of inequality does not affect the symbol of inequality.
\-12x−2y+12x⩾42+12x \-2y⩾12x+42   \- 12x - 2y + 12x \geqslant 42 + 12x \\\ \- 2y \geqslant 12x + 42 \;
Now dividing both sides of the inequality with the coefficient of variable yy i.e. −2 - 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 ⩾ to ⩽\geqslant \,to\, \leqslant .
−2y−2⩽12x+42−2 y⩽−12x2−422 y⩽−6x−21  \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+cy = mx + c
So m=−6 c=−21   m = - 6 \\\ c = - 21 \;
Therefore the required slope-intercept form is y⩽−6x−21y \leqslant - 6x - 21 with slope m=−6m = - 6 and intercept as c=−21c = - 21
So, the correct answer is “ c=−21c = - 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 mm is equal to −1m- \dfrac{1}{m} .
3.when a negative term is added on both sides then also the inequality symbol gets reversed.