Answer

Verified

371.1k+ views

**Hint:**In solving the solution of this question, we will first substitute from both the equations. We will write the equation \[x+y=5\] by putting the term y in the left side of equation and the in the right side of the equation, we will put the term y and the constant term 5. From here, we will take the value of y and put this value of y in the equation \[xy=4\] and find the value of x. And, from there we will find the value of y by putting the value of x in any of the equations. After that, we will put the value of x and y in the term \[{{x}^{3}}+{{y}^{3}}\], then we will get the answer.

**Complete step by step solution:**

Let us solve this question.

In this question, we have given two equations that are \[x+y=5\] and \[xy=4\].

Here, we will first solve the equation \[x+y=5\].

We will take the value of x to the right side of equation \[x+y=5\]

So, the equation \[x+y=5\] can also be written as

\[\Rightarrow y=5-x\]

Now, we will put this value of y (that is \[y=5-x\]) in the equation \[xy=4\].

After putting the value of y in the equation \[xy=4\], then we will get

\[\Rightarrow x\left( 5-x \right)=4\]

We can write the above equation as

\[\Rightarrow 5x-{{x}^{2}}=4\]

We can write the above equation as

\[\Rightarrow {{x}^{2}}-5x+4=0\]

As we know that

If the equation is \[a{{x}^{2}}+bx+c=0\], then the value of x will be

\[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]

Hence, the value of x in the equation \[{{x}^{2}}-5x+4=0\] will be

\[x=\dfrac{-\left( -5 \right)\pm \sqrt{{{\left( -5 \right)}^{2}}-4\times 1\times 4}}{2\times 1}\]

The above can also be written as

\[x=\dfrac{5\pm \sqrt{25-16}}{2}=\dfrac{5\pm \sqrt{9}}{2}=\dfrac{5\pm 3}{2}\]

Hence, the value of x will be \[\dfrac{5+3}{2}\] and \[\dfrac{5-3}{2}\]

Or, we can say that the values of x are 4 and 1.

If we will put the value of x as 4 in the equation \[xy=4\], then the value of y will be 1.

If we will put the value of x as 1 in the equation \[xy=4\], then the value of y will be 4.

For the value of x and y as 4 and 1 respectively, then the value of the term \[{{x}^{3}}+{{y}^{3}}\] will be \[{{4}^{3}}+{{1}^{3}}=65\]

For the value of x and y as 1 and 4 respectively, then the value of the term \[{{x}^{3}}+{{y}^{3}}\] will be \[{{1}^{3}}+{{4}^{3}}=65\]

Hence, we get that the value of the term \[{{x}^{3}}+{{y}^{3}}\] is 65.

**Note:**

We should know how to substitute a variable from the two equations, so that we can solve this type of question easily. We should know that if we have a given a quadratic equation in the form of \[a{{x}^{2}}+bx+c=0\], then the value of x for this type of equation will be \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\].

If we want to cross check if our answer is correct, then we can check the equations \[x+y=5\] and \[xy=4\] are satisfying or not by putting the value of x and y,

If we put the value of x as 4 or 1 and y as 1 or 4 in the equation \[x+y=5\] , then the equation will be like: 5=5, hence it satisfies.

And, if we put the value of x as 4 or 1 and y as 1 or 4 in the equation \[xy=4\], then the equation will be like: 4=4, hence it satisfies both the equation and says that answers are correct.

Recently Updated Pages

What are the Advantages and Disadvantages of Algorithm

How do you write 0125 in scientific notation class 0 maths CBSE

The marks obtained by 50 students of class 10 out of class 11 maths CBSE

You are awaiting your class 10th results Meanwhile class 7 english CBSE

Which one of the following was not the cause of the class 10 social science CBSE

Which one of the following cities is not located on class 10 social science CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Mention the different categories of ministers in the class 10 social science CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Who is the executive head of the Municipal Corporation class 6 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Which monarch called himself as the second Alexander class 10 social science CBSE

Select the word that is correctly spelled a Twelveth class 10 english CBSE

Write an application to the principal requesting five class 10 english CBSE