Question

How do you write \[5x - 3y < 6\] in slope intercept form?

Answer 1

Hint: We will compare the given equation of general form of slope intercept form of a linear equation and then do some simplification to find the correct form. Finally we get the required answer.

Formula used: Addition property of inequality:
If we add any constant to the both of the sides of an inequality then the sign of the inequality does not change.
Subtraction property of inequality:
If we subtract any constant or variable from both the sides of the inequality then the sign of the inequality does not change.
Multiplication property of inequality:
If we multiply both the sides of an inequality by any negative quantity then the sign of the inequality changes to its equivalent opposite sign.
Division property of inequality:
If we divide both the sides of an inequality by any negative quantity then the sign of the inequality changes to its equivalent opposite sign.
General form of slope-intercept of an equation is as following:
\[y = mx + c\], where \[m\] is the slope of the line and \[c\] is the constant .
Now, if it is in inequality form then we can use instead of.

Complete step-by-step solution:
The following inequality is given in the question: \[5x - 3y < 6.\]
Now, take \[3y\]on the R.H.S of the inequality, we get: \[5x < 6 + 3y.\]
Now, take \[6\]on the L.H.S of the inequality, we get: \[5x - 6 < 3y.\]
Using the division property of inequality, divide both the sides by \[3\], we get: \[\dfrac{5}{3}x - \dfrac{6}{3} < y.\]
By doing further simplification and re-arrangements, we get:
\[ \Rightarrow \dfrac{5}{3}x - 2 < y\]
\[ \Rightarrow y > \dfrac{5}{3}x - 2.\]

\[\therefore \]The slope intercept form of the above equation is \[y > \dfrac{5}{3}x - 2.\]

Note: Points to remember:
Slope of a line tells us the nature of a straight line.
Slope of a line is defined as the ratio of rise and run of a line between two different coordinates.
So, we can rewrite it as following:
If any two pints are there \[({x_1},{y_1})\] and \[({x_2},{y_2})\], then the slope of the line should be as following:
\[m = \dfrac{{({y_2} - {y_1})}}{{({x_2} - {x_1})}}.\]