Question

How do you solve by substitution $x = 3 - 3y$ and $4y = x + 11$?

Answer 1

Hint: Choose either of the two equations, say (i), and find the value of one variable, say $x$, in terms of the other, i.e., $y$. Then, substitute the value of $x$, obtained in the other equation, i.e., (ii) to get an equation in $y$ and solve the equation to get the value of $y$. Then, substitute the value of $y$ obtained in the expression for $x$ in terms of $y$ obtained to get the value of $x$. The values of $x$ and $y$ obtained constitute the solution of the given system of two linear equations.

Complete step by step solution:
The given system of equations is
$x = 3 - 3y$…(i)
$4y = x + 11$…(ii)
Choose either of the two equations, say (i), and find the value of one variable, say $x$, in terms of the other, i.e., $y$.
$x = 3 - 3y$
Substitute the value of $x$, obtained in the other equation, i.e., (ii) to get an equation in $y$.
$4y = 3 - 3y + 11$
Solve the equation to get the value of $y$.
$4y + 3y = 14$
Add variables and constant terms together.
$ \Rightarrow 7y = 14$
$ \Rightarrow y = 2$
Substitute the value of $y$ obtained in the expression for $x$ in terms of $y$ obtained to get the value of $x$.
$x = 3 - 3\left( 2 \right)$
$ \Rightarrow x = - 3$
The values of $x$ and $y$ obtained constitute the solution of the given system of two linear equations.
Final solution: Hence, the solution of the given system of equations is $x = - 3$, $y = 2$.

Note: We can also find the solution of a given system by Method of Cross-Multiplication.
System of equations:
$x = 3 - 3y$…(i)
$4y = x + 11$…(ii)
By cross-multiplication, we have
$\dfrac{x}{{\begin{array}{*{20}{c}}
  3&{ - 3} \\
  4&{ - 11}
\end{array}}} = \dfrac{{ - y}}{{\begin{array}{*{20}{c}}
  1&{ - 3} \\
  { - 1}&{ - 11}
\end{array}}} = \dfrac{1}{{\begin{array}{*{20}{c}}
  1&3 \\
  { - 1}&4
\end{array}}}$
$ \Rightarrow \dfrac{x}{{ - 33 + 12}} = \dfrac{{ - y}}{{ - 11 - 3}} = \dfrac{1}{{4 + 3}}$
$ \Rightarrow \dfrac{x}{{ - 21}} = \dfrac{{ - y}}{{ - 14}} = \dfrac{1}{7}$
$ \Rightarrow x = \dfrac{{ - 21}}{7} = - 3$ and $y = \dfrac{{14}}{7} = 2$
Final solution: Hence, the solution of the given system of equations is $x = - 3$, $y = 2$.