Question

# Check if $\left( 5,0 \right)$ is a solution of $2x-5y=10$.

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Hint: To check if a given point is a solution, substitute the coordinates of the point in the equation. If both sides of the equation are equal, then it is a valid solution and if they are unequal then it is not a valid solution. Substitute $\left( 5,0 \right)$ in $2x-5y=10$ to check if both sides are equal.

For any given equation in x and y, a solution means that the values of x and y which satisfy the given equation. Thus, to check if a given point $\left( {{x}_{0}},{{y}_{0}} \right)$ is a solution of the equation $ax+by=c$, substitute $x={{x}_{0}}$ and $y={{y}_{0}}$ in the equation.

Thus, after substitution, the equation becomes $a{{x}_{0}}+b{{y}_{0}}=c$ in which a, b and c are all constants. If this equation is valid on both sides, then the solution $\left( {{x}_{0}},{{y}_{0}} \right)$ is a valid solution.

In the question given, $\left( {{x}_{0}},{{y}_{0}} \right)=\left( 5,0 \right)$ and the equation is $2x-5y=10$.

Substituting the given point $\left( 5,0 \right)$ in the equation $2x-5y=10$, we get the LHS as

\begin{align} & 2\cdot \left( 5 \right)-5\cdot \left( 0 \right) \\ & =10-0 \\ & =10 \\ \end{align}

Thus, the value on the LHS is 10, after substituting the point $\left( 5,0 \right)$. This value is the same as that on the RHS.

Since, $LHS=RHS$, therefore the solution is valid.

Note: The validity of the solution can also be checked from the graph of the equation given. Since the equation is linear, it represents a straight line. A solution of the equation means that the point lies on the given straight line. It may be noted that since there are an infinite number of points on a straight line, thus every straight line has infinitely many solutions. Also, a single equation in 2 variables is different from a system of linear equations in 2 variables. A single equation always has infinite solutions, whereas a system of equations may have either no solution, exactly 1 solution or infinitely many solutions depending on whether the lines are parallel, intersecting, or coincident respectively.