Question

How do you write the equation of a line through point $(4,2);$ slope $ = 3$?

Answer 1

Hint: Here we will use the line equation standard formula. Here we are given the points and the slope values, place in the standard formula and then simplify for the required solution.

Complete step-by-step solution:

The general equation of the straight line:

$y - {y_0} = m(x - {x_0})$      ….. (A)

Given point, $({x_0},{y_0}) = (4,2)$

And $m = 3$

Place the given data in the equation (A)

$y - 2 = 3(x - 2)$

Simplify the above equation:

$ \Rightarrow y - 2 = 3x - 6$

The above equation can be re-written as:

$ \Rightarrow 3x - 6 = y - 2$

Take all the terms on the left hand side of the equation. When move any term from one side to another, the sign of the term also changes. Positive term becomes negative and the negative term becomes positive.

$ \Rightarrow 3x - y - 6 + 2 = 0$

Make pair of like terms in the above equation.

$ \Rightarrow 3x - y\underline { - 6 + 2}  = 0$

When you simplify the terms having positive and the negative sign, you have to actual do subtraction and give sign of bigger number.

$ \Rightarrow 3x - y - 4 = 0$

This is the required solution.

Additional Information: The property the product of the slopes of the two perpendicular lines is always equal to $( - 1)$  whereas slope of two parallel lines are always equal to each other. 

Note: Always remember the standard form of the linear equation, slope and intercept equation as the y intercept depends on the standard equation. Also, know the basic identities to simplify the equation such as product of minus and plus gives negative term and the product of minus and minus gives positive term. Always remember to change the sign of the terms when you move any term from one side to another.