
Given x = 2y + 5 and y = (2x - 3)(x + 9).
How many ordered pairs (x, y) satisfy the system of equations shown above?
A. 0
B. 1
C. 2
D. Infinitely many
Answer
481.2k+ views
Hint: A polynomial of degree n, has at most n roots.
Not all solutions may satisfy the given conditions, and some solutions may be repeated.
In order to solve a system of equations with two variables, we find the expression for one of the variables in terms of the other by using one of the equations and then substitute it in the other equation.
The two solutions of the general quadratic equation , are given by:
(i)If , then there are two distinct solutions.
(ii)If , then there are two solutions, each equal to .
Complete step-by-step answer:
Let us form an equation involving only one of the variables, either x or y.
It is given that x = 2y + 5.
Substituting this value of x in the equation y = (2x - 3)(x + 9), we will get:
⇒ y = [2(2y + 5) - 3][(2y + 5) + 9}]
On removing the inside brackets "()" by performing multiplication, we will get:
⇒ y = [4y + 10 - 3][2y + 5 + 9]
⇒ y = (4y + 7)(2y + 14)
On expanding by multiplying the two terms using the distributive property of multiplication:
⇒
⇒
Comparing this equation with the general equation , we can say that a = 8, b = 69 and c = 98.
Let us calculate the value of by substituting the values of a, b and c:
From 1625 > 0, we will get two distinct values of y, say and .
And, since the degree of x in x = 2y + 5 is one, we will get only one value of x for each value of y, say for and for .
The order pairs satisfying the above equations are therefore, and .
Therefore, the number of ordered pairs which satisfy the given system of equations is two.
Note: "How many values of x?" and "What is the value of x?" are two significantly different questions.
We do not need to calculate the exact values of x and y for answering this question, because it asks "How many (x, y)?" only.
The degree of a polynomial/equation is the highest power of the variables occurring in it.
(xy) has a degree of two, because the variables x and y are multiplied together.
An ordered pair is different from other types of pairs in the sense that the order in which the objects appear in the pair is significant.
So, (a, b) and (b, a) are two different ordered pairs.
Not all solutions may satisfy the given conditions, and some solutions may be repeated.
In order to solve a system of equations with two variables, we find the expression for one of the variables in terms of the other by using one of the equations and then substitute it in the other equation.
The two solutions of the general quadratic equation
(i)If
(ii)If
Complete step-by-step answer:
Let us form an equation involving only one of the variables, either x or y.
It is given that x = 2y + 5.
Substituting this value of x in the equation y = (2x - 3)(x + 9), we will get:
⇒ y = [2(2y + 5) - 3][(2y + 5) + 9}]
On removing the inside brackets "()" by performing multiplication, we will get:
⇒ y = [4y + 10 - 3][2y + 5 + 9]
⇒ y = (4y + 7)(2y + 14)
On expanding by multiplying the two terms using the distributive property of multiplication:
⇒
⇒
Comparing this equation with the general equation
Let us calculate the value of
From 1625 > 0, we will get two distinct values of y, say
And, since the degree of x in x = 2y + 5 is one, we will get only one value of x for each value of y, say
The order pairs satisfying the above equations are therefore,
Therefore, the number of ordered pairs which satisfy the given system of equations is two.
Note: "How many values of x?" and "What is the value of x?" are two significantly different questions.
We do not need to calculate the exact values of x and y for answering this question, because it asks "How many (x, y)?" only.
The degree of a polynomial/equation is the highest power of the variables occurring in it.
(xy) has a degree of two, because the variables x and y are multiplied together.
An ordered pair is different from other types of pairs in the sense that the order in which the objects appear in the pair is significant.
So, (a, b) and (b, a) are two different ordered pairs.
Latest Vedantu courses for you
Grade 8 | CBSE | SCHOOL | English
Vedantu 8 CBSE Pro Course - (2025-26)
School Full course for CBSE students
₹45,300 per year
Recently Updated Pages
Master Class 9 General Knowledge: Engaging Questions & Answers for Success

Master Class 9 English: Engaging Questions & Answers for Success

Master Class 9 Science: Engaging Questions & Answers for Success

Master Class 9 Social Science: Engaging Questions & Answers for Success

Master Class 9 Maths: Engaging Questions & Answers for Success

Class 9 Question and Answer - Your Ultimate Solutions Guide

Trending doubts
Where did Netaji set up the INA headquarters A Yangon class 10 social studies CBSE

A boat goes 24 km upstream and 28 km downstream in class 10 maths CBSE

Why is there a time difference of about 5 hours between class 10 social science CBSE

The British separated Burma Myanmar from India in 1935 class 10 social science CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

What are the public facilities provided by the government? Also explain each facility
