Question

Tell whether the ordered pair is a solution of the given system. (-2,-4) {y=1/2x -3, y=-2x-8}

Answer 1

The ordered pair
`(-2,\, -4)` is indeed a solution to the system:

`\left\lbrace\begin{aligned} & y = (1)/(2)\, x - 3 \\ & y = -2\, x - 8\end{aligned}\right.`.

Explanation:

Consider a system of equations about variables
`x` and
`y`. An ordered pair
`(x_(0),\, y_(0))` (where
`x_(0)` and
`y_(0)` are constant) is a solution to that system if and only if all equations in that system hold after substituting in
`x = x_(0)` and
`y = y_(0)`.

For the system in this question,
`(-2,\, -4)` would be a solution only if both equations in the system hold after replacing all
`x` in equations of the system with
`(-2)` and all
`y` with
`(-4)`.

The
`\texttt{LHS}` of the equation
`y = (1/2)\, x - 3` would become
`(-4)`. The
`\texttt{RHS}` of that equation would become
`(1/2) \, (-2) - 3`. The two sides are indeed equal.

Similarly, the
`\texttt{LHS}` of the equation
`y = -2\, x - 8` would become
`(-4)`. The
`\texttt{RHS}` of that equation would become
`(-2)\, (-2) - 8`. The two sides are indeed equal.

Thus,
`x = (-2)` and
`y = (-4)` simultaneously satisfy both equations of the given system. Therefore, the ordered pair
`(-2,\, -4)` would indeed be a solution to that system.