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How do you solve the system of equations $2x+3y=5$ and $x-2y=6$ ?

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Last updated date: 24th Feb 2024
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IVSAT 2024
Answer
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Hint: we are asked to solve the linear equation $2x+3y=5$ and $x-2y=6$ . To find the solution we will first know what are the possible ways to solve such problems then we use our method to solve. We will find the value of x and y one by one. We will use graphical methods to understand different ways to solve our problem.

Complete step by step answer:
We are given a linear equation as $2x+3y=5$ and $x-2y=6$ . We can see that our equations are linear and it has two variables x and y.
So it is a linear equation in two variables we have to find the solution of these two equations. we know that to find the solution of linear equation in two variable these are various way like –
1. Elimination method
2. Substitution method
3. Cross multiplication method
4. Graphical method
To find the solution we can use any of the following and by each method we will always aim at the same.
We will use elimination methods. In this we will eliminate any one of the variables and then solve for the remaining variable. To eliminate any variable, we will make their coefficient equal by multiplying the equation with the appropriate constant and then we add or subtract as required to eliminate. Once a variable is eliminated then we solve for another variable.
Now we have $2x+3y=5$ ……………………. (1)
And $x-2y=6$ …………………………………… (2)
We can see that coefficient of x in (1) in 2 and coefficient of x in (2) is 1.
Sp we multiply eq (2) by by ‘2’ so we get –
$2x-4y=12$ ………………………. (3)
Now we subtract (1) from (3)
So,
$\begin{align}
  & 2x-4y=12 \\
 & 2x+3y=15 \\
 & -\text{ }-\text{ }- \\
 & \text{ }-7y=7 \\
\end{align}$
So we get $-7y=+7$
So, $y=\dfrac{7}{-7}=-1$
So $y=-1$
Now putting $y=-1$ in eq (1), we get –
$\begin{align}
  & 2x+3\left( -1 \right)=5 \\
 & 2x-3=5 \\
\end{align}$
Adding 3 on both sides, we get –
$2x=8$
Divide both sides by 2 so, we get –
$x=\dfrac{6}{2}=4$
Hence, solution is $x=4$ and $y=-1$

Note:
We can cross check our answer by putting $x=4$ and $y=-1$ in both equations if their value satisfies our problem then it is the correct solution.
So putting $x=4$ and $y=-1$ in eq (1)
$\begin{align}
  & 2x+3y=5 \\
 & 2\left( 4 \right)+3\left( -1 \right)=5 \\
\end{align}$
By simplifying, we get –
$\begin{align}
  & 8-3=5 \\
 & 5=5 \\
\end{align}$
So, this is true.
Now putting $x=4$ and $y=-1$ in $x-2y=6$
$\begin{align}
  & 4-2\left( -1 \right)=6 \\
 & 6=6 \\
\end{align}$
This is true.
So we get verified that –
$x=4$ and $y=-1$ is the correct solution.
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