Question

Which of the following equations represents a line that is parallel to the line with equation $y = - 3x + 4$
A. $6x + 2y = 15$
B. $3x - y = 7$
C. $2x - 3y = 6$
D. $x + 3y = 1$

Answer 1

Hint:
We will find the slope for the given equation. After that, we will find the slopes of the given choices, We know that if two lines are parallel, their slopes must be equal. Hence after finding the slopes for all the choices given, we will compare them with the slope of the given equation, the slope of the choice which matches the slope of the equation would be the final answer.

Complete step by step solution:
According to the question we have the equation of the line as follows
 $y = - 3x + 4$
 $ \Rightarrow 3x + y = 4$
Now we know that the slope of a line of the form
 $ax + by = \lambda $
Where $\lambda $ is a constant is given by,
 $m = \dfrac{{ - a}}{b}$
So, the slope of the given line is
 $m = \dfrac{{ - 3}}{1}$
So we have,
 $ \Rightarrow m = - 3$
Now we check the equations given in the choices
For Choice A: $6x + 2y = 15$
We have slope as
 $ \Rightarrow m = \dfrac{{ - a}}{b}$
On substituting values we get,
 $ \Rightarrow m = \dfrac{{ - 6}}{2}$
On simplification we get,
 $ \Rightarrow m = \dfrac{{ - 3}}{1}$
Hence we have,
 $ \Rightarrow m = - 3$
Since slope is the same as the given solution so this is a parallel line
Choice A is correct
For choice B: $3x - y = 7$
We have slope
 $ \Rightarrow m = \dfrac{{ - a}}{b}$
On substituting the values we get,
 $ \Rightarrow m = \dfrac{{ - 3}}{{ - 1}}$
On simplification we get,
 $ \Rightarrow m = \dfrac{3}{1}$
Hence we have,
 $ \Rightarrow m = 3$
Since slope is NOT the same as for the given equation
So, Choice B is incorrect
For choice C: $2x - 3y = 6$
We have slope
 $ \Rightarrow m = \dfrac{{ - a}}{b}$
On substituting the values we get,
 $ \Rightarrow m = \dfrac{{ - 2}}{{ - 3}}$
On simplification we get,
 $ \Rightarrow m = \dfrac{2}{3}$
Since the slope is NOT the same as for the given equation
So, Choice C is incorrect
For Choice D: $x + 3y = 1$
We have slope
 $ \Rightarrow m = \dfrac{{ - a}}{b}$
On substituting the values we get,
 $ \Rightarrow m = \dfrac{{ - 1}}{3}$
Since the slope is NOT the same as for the given equation.
So, Choice D is incorrect.

Hence, the correct option is A.

Note:
Alternate Solution is,
Put the constant of all equations equal to 0, we get the given equation as,
 $y = - 3x$
 $ \Rightarrow 3x + y = 0$
Now do the same for choices, we get
A. $6x + 2y = 0$
B. $3x - y = 0$
C. $2x - 3y = 0$
D. $x + 3y = 0$
Now if the ratio of any of the two equation eliminates x and y from it, we can say that lines are parallel, for example in the ratio of $6x + 2y = 0$ and $3x + y = 0$ , only \[2:1\] is left, that is, x and y are canceled out (eliminated), Hence $6x + 2y = 0$ would be the final answer.