Question

How do you solve $3y-2x=11$ and $y+2x=9$ using substitution?

Answer 1

Hint: To solve the equations $3y-2x=11$ and $y+2x=9$ using the substitution method, we need to obtain the expression of the variable y in terms of the variable x from the equation $y+2x=9$. For this we have to subtract $2x$ from both sides of this equation. Then the obtained expression of y in terms of x must be substituted into the first equation $3y-2x=11$ to get a linear equation in terms of x. On solving the equation, we will get the value of x. Finally, on substituting the value of x into the expression of y in terms of x, we will get the value of y also.

Complete step-by-step solution:
The equations given are
\[\begin{align}
  & \Rightarrow 3y-2x=11......\left( i \right) \\
 & \Rightarrow y+2x=9........\left( ii \right) \\
\end{align}\]
Considering the second equation, we have
$\Rightarrow y+2x=9$
Subtracting $2x$ from both sides, we get
$\begin{align}
  & \Rightarrow y+2x-2x=9-2x \\
 & \Rightarrow y=9-2x........\left( iii \right) \\
\end{align}$
Substituting (iii) in the equation (i) we get
$\begin{align}
  & \Rightarrow 3\left( 9-2x \right)-2x=11 \\
 & \Rightarrow 27-6x-2x=11 \\
 & \Rightarrow 27-8x=11 \\
\end{align}$
Subtracting $27$ from both sides, we get
$\begin{align}
  & \Rightarrow 27-8x-27=11-27 \\
 & \Rightarrow -8x=-16 \\
\end{align}$
Dividing both sides by $-8$ we get
$\begin{align}
  & \Rightarrow \dfrac{-8x}{-8}=\dfrac{-16}{-8} \\
 & \Rightarrow x=2 \\
\end{align}$
Finally, substituting this in (iii) we get
$\begin{align}
  & \Rightarrow y=9-2\left( 2 \right) \\
 & \Rightarrow y=9-4 \\
 & \Rightarrow y=5 \\
\end{align}$
Hence, we have solved the given equations using substitution and obtained $x=2$ and $y=5$.

Note: For obtaining the expression of one variable in terms of the other, we must select that equation in which the coefficient of at least one variable is equal to one. This is because separating the variable whose coefficient is equal to one is easy and after substitution into the other equation, nothing comes in the denominator, making the calculations easy. Also, if the absolute value of the coefficients of a variable is the same in both the equations, like in the given equations $3y-2x=11$ and $y+2x=9$ the absolute value of the coefficients of x is equal to $2$, we can obtain the expression of $2x$ from one and substitute it into the other equation.