Question

How do you solve $ 4x - 3y = 14 $ and $ y = - 3x + 4? $

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:
 $ 4x - 3y = 14 $ …. (A)
 $ y = - 3x + 4 $ ….. (B)
To use the method of elimination the coefficient of any of the two equations of the same variable should be same.
Multiply equation (B) with using the property of equivalent –
 $ 3y = - 9x + 12 $
The above equation can be re-written as
 $ 9x + 3y = 12 $ …. (C)
Add equation (C) with the equation (A), where the left hand side of the equation (c) is added with the left hand side of the equation (A) and similarly on the right hand side of the equations.
 $ (4x - 3y) + (9x + 3y) = 14 + 12 $
When there is a positive sign outside the bracket then there will be no change in the sign of the terms inside the bracket.
 $ 4x - 3y + 9x + 3y = 26 $
Make the like terms together.
 $ \underline {4x + 9x} + \underline {3y - 3y} = 26 $
Like terms with equal values and opposite signs cancels each other
 $ \Rightarrow 13x = 26 $
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
 $ \Rightarrow x = \dfrac{{26}}{{13}} $
Common factors from the numerator and the denominator cancel each other.
 $ \Rightarrow x = 2 $
Place above value in equation (B)
 $ y = - 3x + 4 $
 $ y = - 3(2) + 4 $
Simplify the above equation –
 $
  y = - 6 + 4 \\
  y = - 2 \;
  $
Therefore, the solutions of the set of equations are – $ (x,y) = (2, - 2) $
So, the correct answer is “ $ (x,y) = (2, - 2) $ ”.

Note: Always remember that when we expand the brackets or open the brackets, the 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-
I.Addition of two positive terms gives the positive term
II.Addition of one negative and positive term, you have to do subtraction and give sign of bigger numbers, whether positive or negative.
III.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.