Question

Solve the following system of equations
1/6x-1/2y=2;1/3x-4/5y=-2

Answer 1

It is easy when we follow the method of substitution and probably the quickest way to obtain the answer. So, let us start without any wait by using LaTeX.


`\begin{bmatrix}(1)/(6)x & - & (1)/(2)y & = & 2 \\ \\ (1)/(3)x & - & (4)/(5)y & = & - 2 \end{bmatrix}`

Isolate the variable of x to proceed further for substitution.


`\bf{(1)/(6)x - (1)/(2)y + (1)/(2)y = 2 + (1)/(2)y}`


`\bf{6 * (1)/(6)x = 6 * 2 + 6 * (1)/(2)y}`


`\bf{(6 * 1)/(6)x = 12 + (6)/(2)y}`


`\bf{x = 12 + 3y}`

Let us substitute back into our equation marked as 2nd equation in this current system, that is:


`\bf{(1)/(3) \Big(12 + 3y \Big) - (4)/(5)y = - 2}`


`\bf{(12)/(3) + (3y)/(3) - (4)/(5)y = - 2}`


`\bf{y + 4 - (4)/(5)y = - 2}`


`\bf{y - (4)/(5)y + 4 - 4 = - 2 - 4}`


`\bf{y - (4)/(5)y = - 6}`


`\bf{y * 5 - (4)/(5)y * 5 = - 6 * 5}`


`\bf{5y - 4y = - 30}`


`\boxed{\mathbf{y = - 30}}`


`\bf{\since \quad y = - 30; \: \: x = 12 + 3y}`


`\mathbf{\therefore \quad x = 12 + 3 (- 30)}`


`\mathbf{x = 12 - 3 * 30}`


`\mathbf{x = 12 - 90}`


`\boxed{\mathbf{x = - 78}}`

Hope it helps.