Question

How do you solve the simultaneous equations $2x + 3y = - 3$ and $3x - 2y = 28?$

Answer 1

Hint: Here we are given two sets of equations and there are two variables in it. So, we will use elimination method to find out the required values for the unknowns. To perform elimination like terms from both the equations should have the same coefficient.

Complete step by step solution:
Take the given expressions:
$2x + 3y = - 3$ …. (A)
$3x - 2y = 28$ …. (B)
To use the method of elimination the coefficient of any of the two equations of the same variable should be same.
Multiply equation (A) with using the property of equivalent and multiply equation (B) with using the property of equivalent
$ \Rightarrow 4x + 6y = - 6$ …. (C)
$ \Rightarrow 9x - 6y = 84$ …. (D)
Add equation (C) with the equation (D), where the left hand side of the equation (C) is added with the left hand side of the equation (D) and similarly on the right hand side of the equations.
\[ \Rightarrow (4x + 6y) + (9x - 6y) = ( - 6) + (84)\]
When there is a positive sign outside the bracket then there is no change in the sign of the terms inside the bracket.
\[ \Rightarrow 4x + 6y + 9x - 6y = 78\]
Make the like terms together.
\[ \Rightarrow \underline {4x + 9x} + \underline {6y - 6y} = 78\]
Like terms with equal values and opposite signs cancels each other. Also, when you subtract a bigger number from the smaller number you have given a sign of the bigger number to the resultant value.
$ \Rightarrow 13x = 78$
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
$ \Rightarrow x = \dfrac{{78}}{{13}}$
Common factors from the numerator and the denominator cancel each other.
$ \Rightarrow x = 6$
Place above value in equation (A) $2x + 3y = - 3$
$ \Rightarrow 2(6) + 3y = - 3$
Simplify the above equation –
$
 \Rightarrow 12 + 3y = - 3 \\
 \Rightarrow 3y = - 3 - 12 \\
 \Rightarrow 3y = - 15 \\
 \Rightarrow y = - \dfrac{{15}}{3} \\
 \Rightarrow y = - 5 \\
 $
Therefore, the solutions of the set of equations are – $(x,y) = (6, - 5)$

Note: Always remember that when we expand the brackets or open the brackets, sign outside the bracket is most important. If there is a positive sign outside the bracket then the values inside the bracket does not change and if there is a negative sign outside the bracket then all the terms inside the bracket changes. Positive terms change to negative and negative term changes to positive.