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# Find the solution in the form $x = a,y = 0$ and $x = 0,y = b$ for the equation $2x + 5y = 10,2x + 3y = 6$ . Is there any common solution?

Last updated date: 20th Jun 2024
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Hint: In this question, we need to determine the solution for the equations $2x + 5y = 10,2x + 3y = 6$ . For this, we will use the arithmetic operations on the given equations and evaluate the solution of the equations.

A solution to the equation are values such that the equation is satisfied.
Take the equation $2x + 5y = 10$ and substituting the values of $x = a,y = 0$ , we will get
$\Rightarrow 2a + 5(0) = 10 \\ \Rightarrow a = 5 \\$
So, $\left( {5,0} \right)$ is one solution of $2x + 5y = 10$ .
Now, put $x = 0,y = b$ in $2x + 5y = 10$ , we will get
$\Rightarrow 2(0) + 5b = 10 \\ \Rightarrow b = 2 \\$
So, $\left( {0,2} \right)$ is another solution to $2x + 5y = 10$
Now, we will find the solution to $2x + 3y = 6$
This time we are putting $x = a,y = 0$ , and $x = 0,y = b$ in one step, we get
$2a + 3(0) = 6{\text{ and }}2(0) + 3b = 6$
On solving the above two equations, we get
$\Rightarrow 2a + 3(0) = 6{\text{ and }}2(0) + 3b = 6 \\ \Rightarrow a = 3{\text{ and }}b = 2 \\$
So $\left( {3,0} \right)$ and $\left( {0,2} \right)$ are solutions of $2x + 3y = 6$
For the common solution, we will check whether the condition $\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$ is satisfied or not.
From equations $2x + 5y = 10,2x + 3y = 6$ , the values of the constants are
${a_1} = 2,{b_1} = 5,{c_1} = 10,{a_2} = 2,{b_2} = 3{\text{ and }}{c_2} = 6$
Now, the ratio of ${a_1}$ and ${a_2}$ is $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{2}{2} = 1$
Similarly, the ratio of ${b_1}$ and ${b_2}$ is $\dfrac{{{b_1}}}{{{b_2}}} = \dfrac{5}{3}$
As, $1 \ne \dfrac{5}{3}$
So, $\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$
So, we can say that the given equations have one common solution.

So, the correct answer is “Option C”.

Note: There are many ways to find the solution of simultaneous linear equations like substitution method, elimination method etc.
We can also find a common solution by plotting equations graphically. If two lines intersect at one point, then they have a unique solution. The condition of the common solution of simultaneous linear equations ${a_1}x + {b_1}y + {c_1} = 0,{a_2}x + {b_2}y + {c_2} = 0$ is $\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$ .