Check if \[\left( 5,0 \right)\] is a solution of \[2x-5y=10\].
Answer
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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.
Complete step-by-step answer:
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.
Complete step-by-step answer:
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.
Last updated date: 21st Sep 2023
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