Which of the following is a characteristic of an inconsistent pair of linear equations?

They have infinitely many solutions.

Their graphs are coincident lines.

Which graph correctly represents a pair of linear equations with no solution?

Graph showing two lines that never meet.

Graph showing two lines intersecting at a point.

Graph showing two lines overlapping each other.

Graph showing two lines that never meet.

Graph showing a single curved line.

Assertion (A): The pair of equations x + 2y - 5 = 0 and -3x - 6y + 15 = 0 has infinitely many solutions. Reason (R): The ratio a1/a2 = b1/b2 = c1/c2 for the given pair of equations.

A and R both correct, R explains A

Assertion (A): The pair of equations 2x + 3y = 7 and 4x + 6y = 10 is inconsistent. Reason (R): Inconsistent pairs of linear equations have parallel lines as their graphical representation.

A and R both correct, R explains A

Analyze the given graph of two linear equations. What can be concluded about the consistency of this pair of equations?

Consistent, with a unique solution

Consistent, with infinitely many solutions

Is this Class 10 Maths Chapter 3 quiz free?

Yes, this quiz on 'Pair of Linear Equations in Two Variables' for CBSE Class 10 Maths is completely free to use, with no hidden costs or sign-ups required.

How many questions are in the quiz?

The main game mode quiz contains 40 questions, covering various difficulty levels from easy to hard. There are also additional book questions and a monthly test.

Is this quiz aligned with the NCERT syllabus?

Absolutely! All questions are carefully designed based on the NCERT Class 10 Maths textbook and syllabus for the 'Pair of Linear Equations in Two Variables' chapter.

What topics does this quiz cover?

This quiz covers graphical methods, algebraic methods (substitution and elimination), conditions for consistency/inconsistency, and determining the nature of solutions (unique, no solution, infinitely many solutions).

Can I use this quiz offline?

Currently, the quiz requires an internet connection to access. However, you can play it on any device with a web browser.

What are consistent and inconsistent pairs of linear equations?

A consistent pair of linear equations has at least one solution (intersecting or coincident lines), while an inconsistent pair has no solution (parallel lines).

Home › CBSE › Class 10 › Maths › PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

CBSE · Class 10 · 🧮 Maths · Chapter 3

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

Graphical method of solving linear equationsConsistent and inconsistent pairs of equationsDependent pair of linear equationsSubstitution methodElimination methodConditions for unique, no, or infinite solutions

Chapter 3, 'Pair of Linear Equations in Two Variables', introduces students to solving systems of two linear equations. It covers graphical methods for visualizing solutions (intersecting, parallel, or coincident lines) and algebraic techniques such as the substitution method and the elimination method. Understanding these concepts is crucial for solving real-world problems and forms a fundamental basis for higher-level mathematics.

Pair of Linear Equations in Two Variables: Basic Concepts

Linear equation in two variables ka general form hota hai: \(ax + by + c = 0\), jahaan \(a, b, c\) real numbers hain aur \(a \neq 0\) ya \(b \neq 0\) (ya dono non-zero). Is chapter mein hum pair of linear equations ki baat karenge, matlab do aise equations ek saath.

General Form of a Pair of Linear Equations:
\(a_1x + b_1y + c_1 = 0\)
\(a_2x + b_2y + c_2 = 0\)

Jahaan \(a_1, b_1, c_1, a_2, b_2, c_2\) real numbers hain aur \(a_1^2 + b_1^2 \neq 0\) aur \(a_2^2 + b_2^2 \neq 0\).

Solution of a Pair of Linear Equations:
\(x\) aur \(y\) ki aisi values jo dono equations ko simultaneously satisfy karein, unhe solution kehte hain.
Geometrically, solution ka matlab hai lines ka intersection point.

Real-life situations mein application:
Bahut saari daily life problems ko linear equations ke form mein represent kiya ja sakta hai. Jaise, cost of items, speed-distance problems, age problems, etc.

Example:
Agar Akhila ne Giant Wheel par \(x\) rides li aur Hoopla \(y\) times khela.
Hoopla = half the number of rides: \(y = \frac{1}{2}x \implies x - 2y = 0\) (Equation 1)
Cost: Ride \(₹3\), Hoopla \(₹4\), Total spent \(₹20\): \(3x + 4y = 20\) (Equation 2)
Ye ban gaya ek pair of linear equations: \(x - 2y = 0\) aur \(3x + 4y - 20 = 0\).

Graphical Representation:
Har linear equation ek straight line represent karta hai.
Pair of linear equations ka matlab hai plane mein do straight lines.
In do lines ke beech teen possibilities ho sakti hain:

Lines intersect at a single point.
Lines are parallel (never intersect).
Lines are coincident (one line lies exactly on the other).

Algebraic Interpretation:
Har graphical case ka ek corresponding algebraic interpretation hota hai terms of solutions.

📖Definition

Linear Equation in Two Variables: An equation of the form \(ax + by + c = 0\), jahaan \(a, b, c\) real numbers hain aur \(a\) aur \(b\) dono zero nahi ho sakte. Iska graph ek straight line hota hai.

❗Important

Har linear equation ek straight line represent karta hai. Pair of linear equations ka solution un lines ka intersection point hota hai.

Graphical Method of Solving a Pair of Linear Equations

Graphical method mein hum dono equations ke liye points plot karke lines draw karte hain. Jahan lines intersect karti hain, wahi solution hota hai.

Steps to solve graphically:

Har equation ke liye kam se kam do solutions find karo. Ye solutions ordered pairs \((x, y)\) ke form mein honge.
In points ko Cartesian plane par plot karo.
Har equation ke liye points ko join karke straight line draw karo.
Observe karo ki lines kaise behave kar rahi hain:

Intersecting Lines: Agar lines ek point par cut karti hain, toh wahi point \((x, y)\) unique solution hai. Is case mein, pair of equations consistent hota hai.
Parallel Lines: Agar lines ek doosre ke parallel hain (kabhi intersect nahi karti), toh no solution. Is case mein, pair of equations inconsistent hota hai.
Coincident Lines: Agar dono lines ek doosre ke upar hain (ek hi line represent karti hain), toh infinitely many solutions. Har point jo line par hai, woh solution hai. Is case mein, pair of equations dependent aur consistent hota hai.

Example (NCERT): Solve \(x - 2y = 0\) and \(3x + 4y - 20 = 0\) graphically.
Equation 1: \(x - 2y = 0 \implies x = 2y\)
Agar \(y=0\), toh \(x=0\) \(\implies (0,0)\)
Agar \(y=1\), toh \(x=2\) \(\implies (2,1)\)
Agar \(y=2\), toh \(x=4\) \(\implies (4,2)\)
Equation 2: \(3x + 4y - 20 = 0 \implies 4y = 20 - 3x \implies y = \frac{20 - 3x}{4}\)
Agar \(x=0\), toh \(y = \frac{20}{4} = 5\) \(\implies (0,5)\)
Agar \(x=4\), toh \(y = \frac{20 - 12}{4} = \frac{8}{4} = 2\) \(\implies (4,2)\)
Jab in points ko plot karke lines draw karenge, toh they will intersect at \((4,2)\). So, unique solution is \(x=4, y=2\).

Limitations of Graphical Method:
Agar solution non-integral coordinates mein ho (e.g., \((1.5, 2.7)\)), toh graph se exact value read karna difficult ho jaata hai.
Isliye, algebraic methods zyada precise hote hain.

💡Tip

Board exams mein, graphical method ke questions mein graph paper use karna mandatory hota hai. Points ko accurately plot karna aur lines ko straight draw karna bahut important hai.

📖Definition

Consistent System: A pair of linear equations jiska at least ek solution ho (unique ya infinitely many). Inconsistent System: A pair of linear equations jiska koi solution na ho. Dependent System: A consistent system jiske infinitely many solutions hon (coincident lines). Dependent systems hamesha consistent hote hain.

Algebraic Method: Substitution Method

Graphical method ki limitations ko overcome karne ke liye, hum algebraic methods use karte hain. Substitution method mein, hum ek equation se ek variable ki value doosre variable ke terms mein nikaalte hain aur use doosre equation mein substitute kar dete hain.

Steps for Substitution Method:

Step 1: Kisi bhi ek equation se (jo convenient lage), ek variable ki value doosre variable ke terms mein express karo. For example, \(y\) ko \(x\) ke terms mein express karo (ya vice versa).

Example: \(x + 2y = 3 \implies x = 3 - 2y\)

Step 2: Is express ki hui value ko doosre equation mein substitute karo. Isse aapko ek linear equation in one variable milega, jise solve karna easy hoga.

Example: Agar doosra equation \(7x - 15y = 2\) hai, toh \(7(3 - 2y) - 15y = 2\).

Step 3: Step 2 se jo variable ki value mili hai (e.g., \(y\)), use Step 1 mein express kiye gaye equation mein substitute karo. Isse aapko doosre variable ki value mil jaayegi (e.g., \(x\)).
Step 4: Apne solutions ko original equations mein check karo ki woh satisfy ho rahe hain ya nahi.

Example (NCERT): Solve \(x + 2y - 4 = 0\) and \(2x + 4y - 12 = 0\) using substitution.
Equation 1: \(x + 2y = 4 \implies x = 4 - 2y\) (Equation 3)
Substitute Equation 3 in Equation 2: \(2(4 - 2y) + 4y - 12 = 0\)
\(8 - 4y + 4y - 12 = 0\)
\(-4 = 0\)
Ye ek false statement hai. Iska matlab hai ki is pair of equations ka koi solution nahi hai. (Graphically, ye parallel lines hongi).

Another Example (NCERT): Solve \(2x + 3y = 9\) and \(4x + 6y = 18\) using substitution.
Equation 1: \(2x + 3y = 9 \implies 2x = 9 - 3y \implies x = \frac{9 - 3y}{2}\) (Equation 3)
Substitute Equation 3 in Equation 2: \(4\left(\frac{9 - 3y}{2}\right) + 6y = 18\)
\(2(9 - 3y) + 6y = 18\)
\(18 - 6y + 6y = 18\)
\(18 = 18\)
Ye ek true statement hai, jo \(y\) ki kisi bhi value ke liye valid hai. Iska matlab hai ki is pair of equations ke infinitely many solutions hain. (Graphically, ye coincident lines hongi).

When to use: Substitution method generally tab convenient hota hai jab kisi ek equation se ek variable ko easily isolate kiya ja sake (e.g., coefficient 1 ho).

❗Important

Substitution method mein, agar substitution ke baad false statement mile (e.g., \(5=0\)), toh no solution. Agar true statement mile (e.g., \(7=7\)), toh infinitely many solutions.

💡Tip

Calculations mein sign errors se bachne ke liye brackets ka sahi use karein jab substitute kar rahe hon.

Algebraic Method: Elimination Method

Elimination method mein, hum ek variable ko eliminate karte hain (hata dete hain) equations ko add ya subtract karke. Iske liye, hum variables ke coefficients ko equal banate hain.

Steps for Elimination Method:

Step 1: Dono equations ko suitable non-zero constants se multiply karo, taaki ek variable ke coefficients numerically equal ho jaayein (sign same ya opposite ho sakte hain).

Example: \(2x + 3y = 8\) aur \(3x + 2y = 7\). Agar \(x\) ko eliminate karna hai, toh pehle equation ko 3 se aur doosre ko 2 se multiply karo. \(6x + 9y = 24\) aur \(6x + 4y = 14\).

Step 2: Ab, equations ko add ya subtract karo, depending on whether the equal coefficients have opposite signs or same signs.

Agar coefficients ke signs opposite hain (e.g., \(5x\) aur \(-5x\)), toh equations ko add karo.
Agar coefficients ke signs same hain (e.g., \(5x\) aur \(5x\)), toh equations ko subtract karo.
Is step se ek variable eliminate ho jaayega, aur aapko ek linear equation in one variable milega.
Example: \((6x + 9y = 24) - (6x + 4y = 14) \implies 5y = 10\).

Step 3: Step 2 se jo variable ki value mili hai (e.g., \(y\)), use kisi bhi original equation mein substitute karo doosre variable ki value find karne ke liye (e.g., \(x\)).

Example: \(5y = 10 \implies y = 2\). Ab \(2x + 3(2) = 8 \implies 2x + 6 = 8 \implies 2x = 2 \implies x = 1\).

Step 4: Apne solutions ko original equations mein check karo.

Example (NCERT): Solve \(x + y = 5\) and \(2x - 3y = 4\) using elimination.
Equation 1: \(x + y = 5\)
Equation 2: \(2x - 3y = 4\)
Step 1: \(y\) ko eliminate karte hain. Equation 1 ko 3 se multiply karo: \(3(x + y) = 3(5) \implies 3x + 3y = 15\) (Equation 3)
Step 2: Equation 3 aur Equation 2 ko add karo (kyunki \(y\) ke coefficients ke signs opposite hain: \(+3y\) aur \(-3y\)).
\((3x + 3y) + (2x - 3y) = 15 + 4\)
\(5x = 19 \implies x = \frac{19}{5}\)
Step 3: \(x = \frac{19}{5}\) ko Equation 1 mein substitute karo:
\(\frac{19}{5} + y = 5 \implies y = 5 - \frac{19}{5} = \frac{25 - 19}{5} = \frac{6}{5}\)
Solution: \(x = \frac{19}{5}, y = \frac{6}{5}\)

When to use: Elimination method generally preferred hota hai jab coefficients integers hon aur easily LCM lekar equal banaye ja sakein. Ye method fractions aur decimals mein bhi effective hai.

🚧Misconception

Multiply karte waqt, poore equation ko multiply karna mat bhoolna (LHS aur RHS dono ko). Sirf ek term ko multiply karne se equation unbalanced ho jaayega.

💡Tip

Elimination method zyada efficient hota hai complex equations ke liye ya jab coefficients bade hon. Practice se speed aur accuracy badhti hai.

Conditions for Solvability and Nature of Solutions

Hum bina equations ko solve kiye bhi unke solutions ka nature bata sakte hain, coefficients ke ratios ko compare karke.

General form:
Equation 1: \(a_1x + b_1y + c_1 = 0\)
Equation 2: \(a_2x + b_2y + c_2 = 0\)

Case 1: Intersecting Lines (Unique Solution)
Condition: \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\)
Graphical: Lines intersect at exactly one point.
Algebraic: Exactly one solution.
System: Consistent.

Case 2: Coincident Lines (Infinitely Many Solutions)
Condition: \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\)
Graphical: Lines overlap completely.
Algebraic: Infinitely many solutions.
System: Dependent and Consistent.

Case 3: Parallel Lines (No Solution)
Condition: \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\)
Graphical: Lines never intersect.
Algebraic: No solution.
System: Inconsistent.

Important Note: \(c_1\) aur \(c_2\) ko compare karte waqt, make sure ki dono equations mein constant terms ya toh dono LHS mein hon ya dono RHS mein hon. Agar ek LHS mein aur doosra RHS mein hai, toh sign change karke unhe same side par le aao ya fir \(c_1/c_2\) ke sign ko adjust karo.

Example: Check the nature of solutions for \(3x + 4y = 10\) and \(6x + 8y = 20\).
Equations ko standard form mein likho: \(3x + 4y - 10 = 0\) aur \(6x + 8y - 20 = 0\).
\(a_1 = 3, b_1 = 4, c_1 = -10\)
\(a_2 = 6, b_2 = 8, c_2 = -20\)
\(\frac{a_1}{a_2} = \frac{3}{6} = \frac{1}{2}\)
\(\frac{b_1}{b_2} = \frac{4}{8} = \frac{1}{2}\)
\(\frac{c_1}{c_2} = \frac{-10}{-20} = \frac{1}{2}\)
Since \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\), the lines are coincident and have infinitely many solutions. The system is dependent and consistent.

⭐Remember

Ye ratio comparison method objective type questions aur quick verification ke liye bahut useful hai. Bina solve kiye aap solution ka nature bata sakte ho.

💡Tip

Agar equations mein constant term RHS par ho, jaise \(a_1x + b_1y = c_1\), toh ratios compare karte waqt \(c_1/c_2\) ko bhi consider karna hai. Agar ek equation mein constant LHS par hai aur doosre mein RHS par, toh ek ko adjust kar lo.

Easy (15)

Which of the following represents a linear equation in two variables?
- 2x + 3 = 0
- x² + y = 5
- 3x + 4y = 7 (correct)
- xy = 10
Why: A linear equation in two variables has the form ax + by + c = 0, where a, b, c are real numbers and a, b are not both zero.
If a pair of linear equations has no solution, what kind of lines do they represent graphically?
- Intersecting lines
- Coincident lines
- Parallel lines (correct)
- Perpendicular lines
Why: Parallel lines never intersect, meaning there is no common point and thus no solution.
What is the nature of a pair of linear equations that has infinitely many solutions?
- Inconsistent
- Consistent and dependent (correct)
- Consistent and independent
- Unique
Why: Infinitely many solutions imply coincident lines, which means the equations are consistent and dependent.
Which algebraic method involves expressing one variable in terms of the other?
- Elimination method
- Substitution method (correct)
- Graphical method
- Cross-multiplication method
Why: The substitution method involves solving one equation for one variable and substituting that expression into the other equation.
For the equations x + 2y = 4 and 2x + 4y = 12, what is the relationship between their ratios a1/a2, b1/b2, and c1/c2?
- a1/a2 ≠ b1/b2
- a1/a2 = b1/b2 = c1/c2
- a1/a2 = b1/b2 ≠ c1/c2 (correct)
- a1/a2 = c1/c2 ≠ b1/b2
Why: For x + 2y - 4 = 0 and 2x + 4y - 12 = 0, a1/a2 = 1/2, b1/b2 = 2/4 = 1/2, and c1/c2 = -4/-12 = 1/3. So, a1/a2 = b1/b2 ≠ c1/c2.
If the lines representing a pair of linear equations intersect at a single point, how many solutions does the pair have?
- No solution
- Infinitely many solutions
- Exactly one solution (correct)
- Two solutions
Why: Intersecting lines meet at exactly one point, indicating a unique solution.
Which of the following is a characteristic of an inconsistent pair of linear equations?
- They have a unique solution.
- They have infinitely many solutions.
- They have no solution. (correct)
- Their graphs are coincident lines.
Why: An inconsistent pair of linear equations has no solution, meaning their graphs are parallel lines.
The equation 2x + 3y = 5 is an example of a linear equation in how many variables?
- One
- Two (correct)
- Three
- Zero
Why: The equation 2x + 3y = 5 contains two distinct variables, x and y.
True or False: The graphical method is always convenient for finding solutions, even with non-integral coordinates.
Why: The graphical method can be inconvenient for non-integral coordinates, as reading exact values can be difficult.
True or False: A dependent pair of linear equations is always consistent.
Why: Dependent pairs have infinitely many solutions, which means they are consistent.
True or False: If a1/a2 ≠ b1/b2 for a pair of linear equations, the lines are parallel.
Why: If a1/a2 ≠ b1/b2, the lines intersect at a unique point, they are not parallel.
The general form of a linear equation in two variables is ax + by + c = ____.
Why: The standard form of a linear equation in two variables is ax + by + c = 0.
A pair of linear equations which has no solution is called an ____ pair of linear equations.
Why: Equations with no solution are termed inconsistent.
The method of solving linear equations by eliminating one variable first is known as the ____ method.
Why: The elimination method focuses on removing one variable to solve for the other.
If a pair of linear equations has a solution, it is called a ____ pair of linear equations.
Why: Any pair of equations with at least one solution is consistent.

Medium (15)

The sum of the digits of a two-digit number is 9. If 27 is added to the number, its digits are reversed. Find the number.
- 27
- 36 (correct)
- 45
- 54
Why: Let the number be 10x + y. Equations are x+y=9 and 10y+x = 10x+y+27. Solving gives x=3, y=6, so the number is 36.
For what value of 'k' will the following pair of linear equations have infinitely many solutions? 2x + 3y = 7 and (k-1)x + (k+2)y = 3k
- k = 7
- k = 5
- k = 10 (correct)
- k = 1
Why: For infinitely many solutions, a1/a2 = b1/b2 = c1/c2. Equating 2/(k-1) = 3/(k+2) and 3/(k+2) = 7/(3k) gives k=10.
A fraction becomes 9/11 if 2 is added to both the numerator and the denominator. If 3 is added to both the numerator and the denominator it becomes 5/6. Find the fraction.
- 5/7
- 7/9 (correct)
- 3/5
- 9/11
Why: Let the fraction be x/y. (x+2)/(y+2)=9/11 and (x+3)/(y+3)=5/6. Solving these gives x=7, y=9.
The difference between two numbers is 26 and one number is three times the other. Find the numbers.
- 13 and 39 (correct)
- 10 and 30
- 26 and 78
- 5 and 15
Why: Let the numbers be x and y. x - y = 26 and x = 3y. Substitute x in the first equation to find y, then x.
Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob's age was seven times that of his son. What are their present ages?
- Jacob: 40, Son: 10 (correct)
- Jacob: 35, Son: 5
- Jacob: 50, Son: 15
- Jacob: 45, Son: 12
Why: Let Jacob's age be x and son's age be y. (x+5)=3(y+5) and (x-5)=7(y-5). Solving gives x=40, y=10.
Match the following conditions for a pair of linear equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 with their graphical representation and algebraic interpretation.
Why: These are the standard conditions for comparing ratios of coefficients to determine the nature of solutions and lines.
Match the following terms with their definitions.
Why: These definitions are crucial for classifying pairs of linear equations based on their solutions.
Match the following pairs of equations with their graphical representation.
Why: Compare ratios a1/a2, b1/b2, c1/c2 for each pair to determine the line type.
The algebraic method that involves adding or subtracting equations to remove one variable is called the ____ method.
Why: Elimination method aims to remove one variable by manipulating equations.
If a pair of linear equations are represented by coincident lines, they have ____ many solutions.
Why: Coincident lines overlap completely, sharing every point as a solution.
Arrange the steps for solving a pair of linear equations using the substitution method in the correct order.
Why: The substitution method involves isolating a variable, substituting it, solving, and then back-substituting.
Arrange the types of graphical representation in increasing order of the number of solutions they yield.
Why: Parallel lines have 0 solutions, intersecting lines have 1 solution, and coincident lines have infinitely many solutions.
Identify the type of lines represented by the pair of equations: x + y = 5 and 2x + 2y = 10.
- Intersecting lines
- Parallel lines
- Coincident lines (correct)
- Perpendicular lines
Why: For x+y=5 and 2x+2y=10, a1/a2 = 1/2, b1/b2 = 1/2, c1/c2 = 5/10 = 1/2. Since all ratios are equal, the lines are coincident.
Which graph correctly represents a pair of linear equations with no solution?
- Graph showing two lines intersecting at a point.
- Graph showing two lines overlapping each other.
- Graph showing two lines that never meet. (correct)
- Graph showing a single curved line.
Why: No solution means the lines are parallel and never intersect.
Observe the graph. What type of solution does this pair of linear equations have?
- No solution
- Unique solution (correct)
- Infinitely many solutions
- Two solutions
Why: The graph shows two lines intersecting at exactly one point, which means a unique solution.

Hard (10)

Solve for x and y: x + y = 14 and x - y = 4. What is the value of x?
Why: Adding the two equations (x+y) + (x-y) = 14+4 gives 2x=18, so x=9.
The cost of 2 pencils and 3 erasers is ₹9. The cost of 4 pencils and 6 erasers is ₹18. If the cost of one pencil is ₹x and one eraser is ₹y, how many solutions does this system of equations have?
Why: The equations are 2x+3y=9 and 4x+6y=18. Comparing ratios a1/a2=2/4=1/2, b1/b2=3/6=1/2, c1/c2=9/18=1/2. Since all ratios are equal, there are infinitely many solutions.
A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. Find the speed of the stream in km/hr.
Why: Let u be speed upstream, v be speed downstream. 30/u + 44/v = 10 and 40/u + 55/v = 13. Solve for u and v, then speed of stream = (v-u)/2.
What is the value of 'a' for which the pair of linear equations 3x + y = 1 and (2a-1)x + (a-1)y = 2a + 1 has no solution?
Why: For no solution, a1/a2 = b1/b2 ≠ c1/c2. Set 3/(2a-1) = 1/(a-1) and solve for a. Then verify the third ratio.
Assertion (A): The pair of equations x + 2y - 5 = 0 and -3x - 6y + 15 = 0 has infinitely many solutions. Reason (R): The ratio a1/a2 = b1/b2 = c1/c2 for the given pair of equations.
Why: Both assertion and reason are correct, and the reason correctly explains why there are infinitely many solutions.
Assertion (A): The pair of equations 2x + 3y = 7 and 4x + 6y = 10 is inconsistent. Reason (R): Inconsistent pairs of linear equations have parallel lines as their graphical representation.
Why: The equations satisfy a1/a2 = b1/b2 ≠ c1/c2, indicating parallel lines and no solution (inconsistent). The reason defines inconsistent pairs correctly.
A shopkeeper sells two types of articles, A and B. The cost of 3 articles of type A and 4 articles of type B is ₹260. The cost of 5 articles of type A and 2 articles of type B is ₹220. If a customer buys 1 article of type A and 1 article of type B, how much will they pay?
- ₹70 (correct)
- ₹80
- ₹60
- ₹90
Why: Let cost of A be x, B be y. 3x+4y=260 and 5x+2y=220. Solve for x and y, then find x+y.
A part of monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for 20 days, she has to pay ₹1000 as hostel charges, whereas a student B, who takes food for 26 days, pays ₹1180. What are the fixed charges?
- ₹180
- ₹400 (correct)
- ₹200
- ₹30
Why: Let fixed charge be x and cost per day be y. x+20y=1000 and x+26y=1180. Solve for x and y.
The graph shows two linear equations. What is the solution to this system?
- (0, 6)
- (6, 0) (correct)
- (0, -4)
- (3, 1)
Why: The lines x+3y=6 and 2x-3y=12 intersect at point B(6,0) on the graph.
Analyze the given graph of two linear equations. What can be concluded about the consistency of this pair of equations?
- Consistent, with a unique solution
- Consistent, with infinitely many solutions
- Inconsistent, with no solution (correct)
- Dependent, with two solutions
Why: The graph shows parallel lines, which means they never intersect and thus have no solution, making the pair inconsistent.