Question

How do you solve the system of equations \[-6x-4y=2\] and \[6x+4y=-6\]?

Answer 1

Hint: From the question given we have been asked to solve \[-6x-4y=2\] and\[6x+4y=-6\]. By observing these two equations, given equations don't intersect at any point because these two equations are parallel lines. By using the concept of slope, we can prove that these two equations are parallel lines.

Complete step by step solution:
We know that if \[y=mx+c\] is an equation then m is the slope of a line.
From the question given, it has been given that \[-6x-4y=2\]
Let us assume this equation be \[\left( 1 \right)\].
\[6x+4y=-6\]
Let us assume this equation be \[\left( 2 \right)\].
First of all, let us solve the second equation, \[6x+4y=-6\]
Shift \[6x\] from the left-hand side of the equation to the right-hand side of the equation.
By shifting \[6x\] from left hand side of the equation to the right-hand side of the equation, we get
\[\Rightarrow 4y=-6x-6\]
Then
\[\Rightarrow y=\dfrac{-6x-6}{4}\]
\[\Rightarrow y=\dfrac{-3x}{2}-\dfrac{3}{2}\]
Therefore, by comparing with the general equation of line.
The slope of the equation \[\left( 2 \right)\]is
\[\Rightarrow {{m}_{1}}=\dfrac{-3}{2}\]
Now, let us solve the first equation, \[-6x-4y=2\]
Shift \[-4y\] from the left-hand side of the equation to the right-hand side of the equation and subtract \[2\] on both sides.
By shifting\[-4y\] from left hand side of the equation to the right-hand side of the equation and subtract \[2\] on both sides, we get
\[\Rightarrow -6x-2=4y\]
Then
\[\Rightarrow y=\dfrac{-6x-2}{4}\]
\[\Rightarrow y=\dfrac{-3x}{2}-\dfrac{1}{2}\]
Therefore, by comparing with the general equation of line.
The slope of the equation\[\left( 1 \right)\] is
\[\Rightarrow {{m}_{2}}=\dfrac{-3}{2}\]
Therefore, the solution for the given equation’s slopes is equal.
\[\Rightarrow {{m}_{1}}={{m}_{2}}=\dfrac{-3}{2}\]

Hence, the given equations are parallel and never intersect.

Note: We should have to observe the equations very carefully while reading the question. For this question we can say that these two lines are parallel by seeing them. Another way to solve can be by subtracting equation 1 with equation 2 we get \[-6x-4y-6x-4y=2+6\] \[\Rightarrow 0=8\] here we can see lhs is not equal to rhs so they won't intersect. While shifting the variables take care of the sign conventions.