Question

How do you solve the linear equation \[2x - y = 3\] and \[2x = 4x - 6\] using the substitution method and are they dependent, independent or inconsistent?

Answer 1

Hint: We have two linear equations with two variables. Since the second equation contains only one variable and simplifying that we will have a value of one variable. To find the other variable we substitute the obtained value in the first equation and simplifying we will have the desired result. Depending on the obtained solution we can tell whether they are dependent, independent or inconsistent.

Complete step-by-step solution:
Given
\[\Rightarrow 2x - y = 3{\text{ }} - - - (1)\]
\[\Rightarrow 2x = 4x - 6{\text{ }} - - - (2)\]
Now take equation (2),
\[\Rightarrow 2x = 4x - 6\]
Now shifting all the ‘x’ term in one side and constant on the other side,
\[\Rightarrow 6 = 4x - 2x\]
\[\Rightarrow 6 = 2x\]
Rearranging we have,
\[\Rightarrow 2x = 6\]
Divide the whole equation by 2
\[\Rightarrow x = \dfrac{6}{2}\]
\[ \Rightarrow x = 3\]
Now to find the other variable ‘y’. We substitute the value of ‘x’ in equation (1).
\[\Rightarrow 2(3) - y = 3\]
\[\Rightarrow 6 - y = 3\]
\[ \Rightarrow - y = 3 - 6\]
\[ \Rightarrow - y = - 3\]
Multiplying ‘-1’ on both side of the equation we have,
\[ \Rightarrow y = 3\]
We can see that the given system of equations is consistent and independent.

Note: We know that the system has a solution hence it is consistent. If a system has no solution it is said to be inconsistent. Also if a consistent system has exactly one solution, it is said to be independent. If a consistent system has an infinite number of solutions it is dependent. We can check whether the obtained solution is correct or not. All we need to do is substitute the obtained solution in the given equations.
\[2(3) - (3) = 3\] and \[2(3) = 4(3) - 6\]
\[6 - 3 = 3\] and \[6 = 12 - 6\]
\[ \Rightarrow 3 = 3\] and \[6 = 6\]. Thus both the equations satisfy. Hence the obtained answer is correct.