Question

How do you solve the system of equations $ 8x + y = - 16 $ and $ - 3x + y = - 5 $ algebraically?

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.

Complete step by step solution:
Take the given expressions:
 $ \Rightarrow 8x + y = - 16 $ …. (A)
 $ \Rightarrow - 3x + y = - 5 $ ….. (B)
To use the method of elimination the coefficient of any of the two equations of the same variable should be the same and here we have a coefficient of “b” are same. Therefore, 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 equation.
 $ \Rightarrow (8x + y) - ( - 3x + y) = - 16 - ( - 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 8x + y + 3x - y = - 16 + 5 $
Make the like terms together.
 $ \Rightarrow \underline {8x + 3x} + \underline {y - y} = - 11 $
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 11x = - 11 $
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
 $ \Rightarrow x = - \dfrac{{11}}{{11}} $
Common factors from the numerator and the denominator cancel each other.
 $ \Rightarrow x = 1 $
Place above value in equation (A)
 $ \Rightarrow 8x + y = - 16 $
 $ \Rightarrow 8( - 1) + y = - 16 $
Simplify the above equation –
 $ \Rightarrow - 8 + y = - 16 $
Move constants on one side –
 $ \Rightarrow y = - 16 + 8 $
 $ \Rightarrow y = - 8 $

Therefore, the solutions of the set of equations are – $ (x,y) = ( - 1, - 8) $

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.

While doing simplification remember the golden rules-
- Addition of two positive terms gives the positive term
- Addition of one negative and positive term, you have to do subtraction and give signs of bigger numbers whether positive or negative.
- Addition of two negative numbers gives a negative number but in actual you have to add both the numbers and give a negative sign to the resultant answer.