Question

Find out the slope-intercept of the following equation \[3x - y = 14\].

Answer 1

Hint: In the given question we are asked to write the slope-intercept form of the given equation of a line. For this, we will keep coordinate y on the left side and move everything else to the right side. After that, we will compare the equation with the standard form of the slope-intercept equation and determine the slope and y-intercept of the equation.

Complete step by step solution:
 We know that in two-dimensional geometry, the slope-intercept form of the equation is given by \[y = mx + c\], where m is the slope of the line and c represents the y-intercept of the line.
If theta is the inclination of any line L then tanθ is called the slope or gradient of the line L. The slope of a line is determined by m. Thus, m= tanθ, θ is not equal to 90°. It is important to note here that the slope of the x-axis is zero and the slope of the y-axis is not defined.
Now, the equation of the line given in the question is, \[3x - y = 14 - - - - - \left( 1 \right)\]
Subtracting \[3x\] from both sides, we get,
\[ \Rightarrow 3x - y - 3x = 14 - 3x\]
\[ \Rightarrow - y = 14 - 3x\]
Multiplying both sides with \[\left( { - 1} \right)\]
We get,
\[ \Rightarrow \left( { - 1} \right) - y = \left( { - 1} \right)\left( {14 - 3x} \right)\]
\[ \Rightarrow y = - 14 + 3x\]
\[ \Rightarrow y = 3x - 14 - - - - - \left( 2 \right)\]
Which is the required slope-intercept form of the equation of the given line.
Further, comparing equation (2) with the standard slope-intercept form of the equation, i.e., \[y = mx + c\]

Therefore, we get, \[m = 3\] and the y-intercept as \[14\].

Note: It is important to note here that there are various forms of a line. They are as follows:
-One point-slope form i.e., \[y - {y_1} = m(x - {x_1})\]
-Two point-slope forms i.e., \[y - {y_1} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}(x - {x_2})\]
-Slope intercept form i.e., \[y = mx + c\]