Question

How do you find the slope and y intercept to sketch given \[7x + 6y = 30\]?

Answer 1

Hint: Rewrite the given line equation to standard line equation \[y = mx + c\] and then compare the terms to find the slope of the given equation, where m is the slope and c is the y intercept.

Complete step by step solution:
Given the line equation,
\[7x + 6y = 30\]
On rewriting the above equation to the standard from of line equation which is
\[y = mx + c\]
Where m is slope here and c is constant
\[ \Rightarrow 7x + 6y = 30\]
Moving x term to right hand side we will get
\[ \Rightarrow 6y = 30 - 7x\]
Now dividing with 6 on both side we will get
\[ \Rightarrow y = \dfrac{{30}}{6} - \dfrac{{7x}}{6}\]
On simplifying the equation we will get
\[
   \Rightarrow y = 5 - \dfrac{7}{6}x \\
   \Rightarrow y = - \dfrac{7}{6}x + 5 \\
 \]
Now on comparing the above line equation with the standard one we will get
\[ \Rightarrow y = mx + c\]
Here\[m = - \dfrac{7}{6},c = 5\].

Therefore the slope of the given equation is\[m = - \dfrac{7}{6}\] and y intercept is \[c = 5\].

Additional information: If you are provided with the line equation, then it is not a big deal to find the slope you can easily find the slope, but when you are provided with two points to find the slope then you have to use the formula
Slope \[ = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\]
Even you can find the y intercept by substituting \[x = 0\] for the given line.

Note: For a line making acute angle with the x-axis, the slope is positive as the behaviour of $y$ is same as that of $x$, i.e. the value of $y$ increases for increase in the value of $x$ and the value of $y$ decreases for decrease in the value of $x$. We can also find the y intercept of the line by putting $x = 0$ in the equation as when the line is cutting the y-axis the value of $x$ is $0$.