Question

Find the slope of the line parallel to the equation \[y + 2x = 2\]
(A) $ - 2$
(B) $2$
(C) $1$
(D) $ - 1$

Answer 1

Hint: Express the given equation in the form $Ax + By + C = 0$ and find its slope using the formula $m = \dfrac{{ - A}}{B}$ where m represents the slope of the given equation.
Use the fact that parallel lines have the same slope to arrive at the required answer.

Complete step by step answer:
Given equation is $y + 2x = 2$
It consists of two variables $x$ and $y$
We will compare this equation with the general equation of a straight line in two variables $x$ and $y$
The general equation of a straight line is given by $Ax + By + C = 0$
Here, the letters A, B, and C are real numbers. Also, A and B are non-zero constants.
Now, the slope of the straight line given by the equation $Ax + By + C = 0$ is $m = \dfrac{{ - A}}{B}$ where $B \ne 0$
We will rewrite the given equation in the form of $Ax + By + C = 0$ to obtain the slope.
Thus, we have the equation $2x + y - 2 = 0$ where the values of A, B, and C are 2, 1, and -2 respectively.
Thus, the slope of $2x + y - 2 = 0$ is $m = \dfrac{{ - 2}}{1} = - 2$
But, the question is not about finding the slope of the equation $2x + y - 2 = 0$ or $y + 2x = 2$
We are asked to find the slope of a line which is parallel to the line represented by $y + 2x = 2$
Now, we know that the general equation of any line which is parallel to the line represented by the equation $Ax + By + C = 0$would be of the form $Ax + By + {C_1} = 0$ where $C \ne {C_1}$ and ${C_1}$ is also a real number.
We can notice here that the coefficients A and B of the variables $x$ and $y$ are the same for a pair of parallel lines.
Therefore, the slope of a line parallel to the line represented by the equation $Ax + By + C = 0$ will also be given by $m = \dfrac{{ - A}}{B}$ where $B \ne 0$
Thus, we can conclude that parallel lines have the same slope.
This helps us to arrive at the answer to our question.
That is, the slope of any line which is parallel to the line represented by the equation $y + 2x = 2$ will be $ - 2$

Hence $ - 2$ is the slope of the line parallel to the equation \[y + 2x = 2\]

Note: A common mistake made by many students is that they tend to take the coefficient of $x$ in the numerator and that of $y$ in the denominator to calculate the slope. One needs to be careful with this substitution which is the key to answering such questions.