Question

Solve the following simultaneous linear equation:
\[2(x - 4) = 9y + 2\], \[x - 6y = 2\]

Answer 1

Hint: Here we have a system of two linear equations with two variables. We need to find the value of ‘x’ and ‘y’. There are many methods to solve this. We are going to use the substitution method for solving this problem. First, we need to solve one equation for one of the variables and then we need to substitute this expression into another equation and we solve it. Using this we will have one variable value and to find the other we substitute the obtained variable value in any one of the given equations.

Complete step-by-step solution:
Given,
\[2(x - 4) = 9y + 2{\text{ }} - - - (1)\]
\[x - 6y = 2{\text{ }} - - - (2)\]
From equation (2) we have,
\[x = 2 + 6y\]
Now we substitute this ‘x’ value in equation (1) we have,
\[2((2 + 6y) - 4) = 9y + 2\]
Thus, we have a linear equation with one variable and we can simplify for ‘y’,
\[2(2 + 6y - 4) = 9y + 2\]
\[2(6y - 2) = 9y + 2\]
\[12y - 4 = 9y + 2\]
Now shifting the variables one side and constant other sides we have,
\[12y - 9y = 2 + 4\]
\[3y = 6\]
Divide the whole equation by 3 we have,
\[y = \dfrac{6}{3}\]
\[ \Rightarrow y = 2\].
To find the value of ‘x’ we need to substitute the obtained ‘y’ value in any one of the equations. Let’s substitute the value of ‘y’ in equation (2) we have,
\[x - 6(2) = 2\]
\[x - 12 = 2\]
\[ \Rightarrow x = 14\].
Thus we have the solution is \[ \Rightarrow x = 14\] and \[ \Rightarrow y = 2\].

Note: To check whether the obtained answer is correct or not, we substitute the obtained value in the given problem.
\[x - 6y = 2\]
\[14 - 6\left( 2 \right) = 2\]
\[14 - 12 = 2\]
\[ \Rightarrow 2 = 2\]
That is equation 2 satisfies, similarly, it will also satisfy equation (3). Hence the obtained answer is correct. We can also solve the given problem by using the elimination method or by cross-product method.