Question

Solve \[4x + 5y = 11{\text{ and }}y = 3x - 13\]

Answer 1

Hint: So we have been given a pair of linear equations which can be solved either by graphical or algebraic method. Here let’s use the algebraic method.
The algebraic method is of three types:
1. Substitution Method
2. Elimination Method
3. Cross Multiplication Method
So by using any of the above methods we can solve the given pair of linear equations.

Complete step by step solution:
Given
\[
  4x + 5y = 11.................................\left( i \right) \\
  y = 3x - 13..........................\left( {ii} \right) \\
 \]
Now since we are using an algebraic method let’s use the substitution method given above to solve the question.
Substitution Method:
In this method from the given two equations of two variables, we have to substitute the equation of any one variable from one of the equations and then substitute it in the other one such that the second equation becomes an equation of one variable and thereby we can solve for that one variable.
So by using the above definition, let’s substitute ‘y’ from equation (ii) to (i):
$ \Rightarrow y = 3x - 13..................\left( {iii} \right)$
Now substituting (iii) in (ii):
$ \Rightarrow 4x + 5y = 11$
Now substituting (iii), we get:
\[
   \Rightarrow 4x + 5\left( {3x - 13} \right) = 11 \\
   \Rightarrow 4x + 15x - 65 = 11 \\
   \Rightarrow 19x - 65 = 11..................\left( {iv} \right) \\
 \]
On observing (iv) we get that it’s an equation of only ‘x’ such that we can solve for the variable ‘x’.
$
   \Rightarrow 19x - 65 = 11 \\
   \Rightarrow 19x = 65 + 11 \\
   \Rightarrow 19x = 76 \\
   \Rightarrow x = \dfrac{{76}}{{19}} \\
   \Rightarrow x = \dfrac{{19 \times 4}}{{19 \times 1}} \\
   \Rightarrow x = 4......................\left( v \right) \\
 $
So now we get:$x = 4$
Now substituting (v) in (iii) to get the value of ‘y’:
\[
   \Rightarrow y = 3x - 13 \\
   \Rightarrow y = 3\left( 4 \right) - 13 \\
   \Rightarrow y = 12 - 13 \\
   \Rightarrow y = - 1.........................\left( {vi} \right) \\
 \]
Therefore on solving \[4x + 5y = 11{\text{ and }}y = 3x - 13\] algebraically we get $x = 4\;\;{\text{and}}\;\;{\text{y}} = - 1.$

Note: The key point to solve this type of equation is to combine all the like terms i.e., finding out the common term and evaluate for the variable asked. As we know that Simultaneous equations are two equations, each with the same two unknowns and are "simultaneous" because they are solved together.