Find the best tutors and institutes for Class 10 Tuition
Q3(v):
On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the following pair of linear equations are consistent, or inconsistent. (v) $\frac{4}{3}x + 2y = 8$ ; 2x + 3y = 12
Solution :
Given: A pair of linear equations in two variables:
Equation 1: $\frac{4}{3}x + 2y = 8$
Equation 2: $2x + 3y = 12$
To Find: Determine whether the given pair of linear equations is consistent or inconsistent by comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$.
Step 1: Standardizing the Equations
The standard form of a linear equation in two variables is $ax + by + c = 0$ or $ax + by = c$. We will express both equations in the form $ax + by = c$.
Equation 1: $\frac{4}{3}x + 2y = 8$
Equation 2: $2x + 3y = 12$
Step 2: Identifying Coefficients
Comparing the given equations with the general form $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$:
For Equation 1: $a_1 = \frac{4}{3}$, $b_1 = 2$, $c_1 = 8$
For Equation 2: $a_2 = 2$, $b_2 = 3$, $c_2 = 12$
Step 3: Calculating the Ratios
Now, we calculate the ratios of the coefficients:
Ratio of $x$-coefficients: $\frac{a_1}{a_2} = \frac{4/3}{2} = \frac{4}{3 \times 2} = \frac{4}{6} = \frac{2}{3}$
Ratio of $y$-coefficients: $\frac{b_1}{b_2} = \frac{2}{3}$
Ratio of constants: $\frac{c_1}{c_2} = \frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3}$
Step 4: Comparing the Ratios and Determining Consistency
We observe that:
$\frac{a_1}{a_2} = \frac{2}{3}$
$\frac{b_1}{b_2} = \frac{2}{3}$
$\frac{c_1}{c_2} = \frac{2}{3}$
Therefore, $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$.
[Justification: According to the algebraic condition for consistency, if $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, the lines are coincident, meaning they have infinitely many solutions. A system of linear equations is defined as consistent if it has at least one solution.]
Since the ratios are equal, the pair of linear equations represents coincident lines and possesses infinitely many solutions.
Final Answer: The pair of linear equations is consistent.
More Questions from Class 10 Mathematics Pair of linear equations in two variable EXERCISE 3.1
- Q1(i): Form the pair of linear equations in the following problems, and find their solutions graphically. (i) 10 students of Class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.
- Q1(ii): Form the pair of linear equations in the following problems, and find their solutions graphically. (ii) 5 pencils and 7 pens together cost ` 50, whereas 7 pencils and 5 pens together cost ` 46. Find the cost of one pencil and that of one pen.
- Q2(i): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (i) 5x – 4y + 8 = 0; 7x + 6y – 9 = 0
- Q2(ii): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (ii) 9x + 3y + 12 = 0; 18x + 6y + 24 = 0
- Q2(iii): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (iii) 6x – 3y + 10 = 0; 2x – y + 9 = 0
- Q3(i): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the following pair of linear equations are consistent, or inconsistent. (i) 3x + 2y = 5 ; 2x – 3y = 7
- Q3(ii): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the following pair of linear equations are consistent, or inconsistent. (ii) 2x – 3y = 8 ; 4x – 6y = 9
- Q3(iii): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the following pair of linear equations are consistent, or inconsistent. (iii) $\frac{3}{2}x + \frac{5}{3}y = 7$ ; 9x – 10y = 14
- Q3(iv): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the following pair of linear equations are consistent, or inconsistent. (iv) 5x – 3y = 11 ; – 10x + 6y = –22
- Q4(i): Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: (i) x + y = 5, 2x + 2y = 10
- Q4(ii): Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: (ii) x – y = 8, 3x – 3y = 16
- Q4(iii): Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: (iii) 2x + y – 6 = 0, 4x – 2y – 4 = 0
- Q4(iv): Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: (iv) 2x – 2y – 2 = 0, 4x – 4y – 5 = 0
- Q5: Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.
- Q6(i): Given the linear equation 2x + 3y – 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is: (i) intersecting lines
- Q6(ii): Given the linear equation 2x + 3y – 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is: (ii) parallel lines
- Q6(iii): Given the linear equation 2x + 3y – 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is: (iii) coincident lines
- Q7: Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region.
CBSE Solutions for Class 10 Mathematics Pair of linear equations in two variable
Chapters in CBSE - Class 10 Mathematics
Top Tutors who teach Pair of linear equations in two variable
Top Tutor
Sector-18, Faridabad
2 yrs of
Exp
5 students
1038 hrs of classes completed
600
per hour
Online Classes
I have approximately 3 years of experience of teaching Mathematics across different boards including CBSE, IGCSE and state board in reputed schools like DPS International. I am post graduate in statistics and honors degree in mathematics from reputed universities like DU. I have helped many students to overcome their fear of Mathematics and Statistics. My aim is to develop a genuine love for the subject in students.
The best math teacher my son has ever had! She explains every concept with clarity, makes learning enjoyable, and ensures he fully understands each topic. Truly inspiring and supportive.
Top Tutor
SH 74, Doddaballapura
Less than a yr
Exp
3 students
540 hrs of classes completed
650
per hour
Online Classes
Your Hard Work. Their Future. Our Responsibility. At Fundamend Academy, we know that every penny you spend on tuition is a penny earned through sacrifice. You aren't just looking for "classes"—you are looking for a teacher who cares as much as you do. Why Fundamend Academy? We’ve removed the risks that usually waste your money and time: Money-Back Peace of Mind: If your child doesn't "click" with their teacher, we swap the tutor immediately at zero extra cost. Your fees will never go to waste. (1) Zero Hidden Costs: All study materials, board exam test series, and session recordings are included. No extra books to buy. (2) Total Honesty: No fake promises. You get a weekly report and a monthly call to discuss your child’s real progress. (3) The Comfort of Home Language: We teach in your child’s native tongue so they finally understand the concept instead of just memorizing it. We don’t just teach; we take the burden off your shoulders so you can see your child succeed. BOOK A FREE DEMO See our teaching for yourself before you spend a single rupee.
I thank from my bottom of my heart to Girish Sir for identifying the gaps in my son, now he secured excellent marks in his boards. Thanks for all the tutors. I can't forget this. I strongly recommend my friends, relatives .
Top Tutor
Begur Road, Bangalore
5 yrs of
Exp
3 students
4 hrs of classes completed
300
per hour
Online Classes
Tutor's home
I have been teaching since last more than 5 years. My way of teaching style is so simple so that it make students very easier to understand. I am taking classes of Math, Hindi and Sanskrit subject.
Top Tutor
Chinsurah R S, Hooghly
15 yrs of
Exp
7 students
987 hrs of classes completed
350
per hour
Online Classes
Tutor's home
I've taught at least 150 students of this Class till date ....online/offline together.
Top Tutor
Tilak Nagar, Delhi
4 yrs of
Exp
2 students
56 hrs of classes completed
300
per hour
Online Classes
Student's Home
Tutor's home
I am a mathematics expert . I am teaching this subject since a decade . I ve handled thousands of students and bring numerous centum results in past . You will able to learn maths in the most pragmatic approach with me and develop a love for this subject .
Find more Tutor for Pair of linear equations in two variable in your City
- Bangalore Mathematics Tutors
- Delhi Mathematics Tutors
- Chennai Mathematics Tutors
- Gurgaon Mathematics Tutors
- Noida Mathematics Tutors
- Hyderabad Mathematics Tutors
- Mumbai Mathematics Tutors
- Ghaziabad Mathematics Tutors
- Chandigarh Mathematics Tutors
- Pune Mathematics Tutors
- Jaipur Mathematics Tutors
- Surat Mathematics Tutors
Download free CBSE - Class 10 Mathematics Pair of linear equations in two variable EXERCISE 3.1 worksheets
Download Now
