Question

How do you solve $ {x^2} + {y^2} = 153,{\text{ y = - 4x?}} $

Answer 1

Hint: Here we are given two equations both have different powers of “x” and “y” so will convert both the powers of variables in the same power format and then will use the substitution method to simplify for the required values.

Complete step by step solution:
Take the given expressions:
 $ \Rightarrow {x^2} + {y^2} = 153 $ …. (A)
 $ \Rightarrow y = - 4x $ …. (B)
Take the square of the above expression –
 $ \Rightarrow {y^2} = 16{x^2} $ ….. (C)
Place the above value in the equation (A)
 $ \Rightarrow {x^2} + 16{x^2} = 153 $
Simplify the above equation –
 $ \Rightarrow 17{x^2} = 153 $
Term multiplicative on one side if moved to the opposite side, then it goes to the denominator.
 $ \Rightarrow {x^2} = \dfrac{{153}}{{17}} $
Find factors for the numerator on the right hand side of the equation.
 $ \Rightarrow {x^2} = \dfrac{{17 \times 9}}{{17}} $
Common factors from the numerator and the denominator cancel each other. Therefore, remove from the numerator and denominator in the above equation.
 $ \Rightarrow {x^2} = 9 $
Take square-root on both the sides of the equation –
 $ \Rightarrow \sqrt {{x^2}} = \sqrt 9 $
Square and square-root cancel each other on the left hand side of the equation –
 $ \Rightarrow x = \pm 3 $ ….(D)
Place the above equation in equation (B) $ y = - 4x $
When, $ x = 3 $ then $ y = - 4(3) = - 12 $ and
When, $ x = - 3 $ then $ y = - 4( - 3) = 12 $ and

Note: Be careful about the sign convention, when you move any term from one side to the other the sign of the term also changes. Positive term becomes negative and negative term becomes positive. Also, remember the square of positive and the negative term gives a result always as positive and square root of the positive term can give negative or positive terms. Be good in square and square-root concepts and remember at least till twenty.