Question

How do you solve the system of equations $x + y = 3$ and $x = 3y - 5?$

Answer 1

Hint: Take the given set of equations and then will use the elimination method which eliminates one variable and gets the equation where one of the values for the variable is known and accordingly can calculate the other value.

Complete step by step solution:
Take the given expressions:
$\Rightarrow x + y = 3$ …. (A)
$\Rightarrow x = 3y - 5$
Take all variables on one side of the equation, when you move any term from one side to another then the sign of the terms also changes. Positive term changes to negative.
 $\Rightarrow x - 3y = - 5$ ….. (B)
To use the method of elimination the coefficient of any of the two equations of the same variable should be same.
Subtract equation (B) from the equation (A), where the left hand side of the equation (B) is subtracted from the left hand side of the equation (A) and similarly on the right hand side of the equations.
$\Rightarrow (x + y) - (x - 3y) = 3 - ( - 5)$
When there is a negative sign outside the bracket then the sign of the terms also changes. Positive terms become negative and vice-versa.
$\Rightarrow x + y - x + 3y = 3 + 5$
Make the like terms together.
\[\Rightarrow \underline {x - x} + \underline {y + 3y} = 8\]
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 4y = 8$
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
$ \Rightarrow y = \dfrac{8}{2}$
Common factors from the numerator and the denominator cancels each other.
$ \Rightarrow y = 4$
Place above value in equation (A)
$x + 4 = 3$
Simplify the above equation –
$
  \Rightarrow x = 3 - 4 \\
 \Rightarrow x = ( - 1) \\
 $
Therefore, the solutions of the set of equations are – $(x,y) = ( - 1,4)$

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.