Question

Solve the following system of equations:
$\begin{align}
  & 3x-7y+10=0 \\
 & y-2x-3=0 \\
\end{align}$

Answer 1

Hint: Here, we have to apply the substitution method to find a solution of the system of equations. First we have to write the second equation in terms of x. After substituting, we will get the equation in one variable x and then solve for x. After finding x, substitute its value in any one of the equations to obtain the value of y. These values are the solutions of the system of equations.

Complete step-by-step answer:
Here, given the system of equations:
$3x-7y+10=0$ …… (1)
$y-2x-3=0$ ……. (2)
Now, we have to find the solution to the above equations.
Here, we can find the solution by substitution method.
From equation (2) we can take -2x and -3 to the right side, which becomes 2x and 3,
$\Rightarrow y=2x+3$
Now, by substituting $y=2x+3$ in equation (1), we obtain:
$\begin{align}
  & 3x-7\times (2x+3)+10=0 \\
 & \Rightarrow 3x-7\times 2x-7\times 3+10=0 \\
 & \Rightarrow 3x-14x-21+10=0 \\
\end{align}$
Next, adding the similar terms,
$\Rightarrow -11x-11=0$
Now, by taking -11 to the right side, it becomes 11,
$\Rightarrow -11x=11$
In the next step, by cross multiplying,
$x=\dfrac{11}{-11}$
Now, by cancellation, we obtain:
$x=-1$
Now by putting $x=-1$ in $y=2x+3$, we obtain the equation:
$\begin{align}
  & \Rightarrow y=2\times -1+3 \\
 & \Rightarrow y=-2+3 \\
 & \Rightarrow y=1 \\
\end{align}$
Hence, we got the values as $x=-1$ and $y=1$.
Therefore, we can say that the solution of the system of equations is $x=-1$ and $y=1$.

Note: Here, we are using a substitution method to solve the system of equations. When using the substitution method, we use the fact that if two expressions y and x are equal value, x = y, then x may replace y or vice-versa in another expression without changing the value of the expression. The elimination method also requires us to add or subtract the equations in order to eliminate either x or y. Often one may not proceed with the addition directly without first multiplying either the first or second equation by some value.