Kaliyuga Ekalavya

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Question: On comparing the ratio, \( \left(\frac{a_1}{a_2}\right), \left(\frac{b_1}{b_2}\right), \left(\frac{c_1}{c_2}\right) \), find out whether the following pair of linear equations are consistent, or inconsistent. \( \frac{4}{3}x + 2y = 8 \) ; \( 2x + 3y = 12 \) Given: The two linear equations are: (1) \( \frac{4}{3}x + 2y = 8 \) (2) \( 2x + 3y = 12 \) To Find: Whether the given pair of linear equations are consistent or inconsistent by comparing the ratios: \( \left(\frac{a_1}{a_2}\right), \left(\frac{b_1}{b_2}\right), \left(\frac{c_1}{c_2}\right) \) Formula: For two linear equations in the form: \( a_1x + b_1y + c_1 = 0 \) \( a_2x + b_2y + c_2 = 0 \) The consistency is checked as follows: Case 1: If \( \frac{a_1}{a_2} \ne \frac{b_1}{b_2} \), equations are consistent and have a unique solution. Case 2: If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2} \), equations a...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Question: On comparing the ratios a 1 , b 1 , c 1 and a 2 , b 2 , c 2 , find out whether the following pair of linear equations are consistent or inconsistent: \(2x - 3y = 8;\quad 4x - 6y = 9.\) Given: \(2x - 3y = 8\) \(4x - 6y = 9\) To Find: Whether the given pair of linear equations is consistent or inconsistent by comparing the ratios of coefficients. Formula: For a pair of linear equations: \(a_1x + b_1y + c_1 = 0\) \(a_2x + b_2y + c_2 = 0\) Consistent & Unique Solution: if \(\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}\) Consistent & Infinitely Many Solutions: if \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}\) Inconsistent: if \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}\) Solution: Step 1: Write both equations in standard form. $$ \begin{align*} 2x - 3y &= 8 \implies 2x - 3y - 8 = 0 \\ 4x - 6...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.1 Question 3(iv) Question: On comparing the ratios a 1 , b 1 , c 1 and a 2 , b 2 , c 2 , find out whether the following pair of linear equations are consistent or inconsistent. \( 5x - 3y = 11 \) ; \( -10x + 6y = -22 \) Given: The pair of linear equations: 1) \( 5x - 3y = 11 \) 2) \( -10x + 6y = -22 \) To Find: Whether the given pair of equations is consistent or inconsistent by comparing the ratios: \[ \frac{a_1}{a_2},\quad \frac{b_1}{b_2},\quad \frac{c_1}{c_2} \] Formula: For two linear equations in the form: \( a_1x + b_1y + c_1 = 0 \) \( a_2x + b_2y + c_2 = 0 \) Compare the ratios: \[ \frac{a_1}{a_2},\quad \frac{b_1}{b_2},\quad \frac{c_1}{c_2} \] Cases: Case 1: If \( \frac{a_1}{a_2} \ne \frac{b_1}{b_2} \), the system is consistent (has a unique solution). Case 2: If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2} \), the system is inconsist...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.1 Question 4(i) Question: Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: \( x + y = 5 \), \( 2x + 2y = 10 \) Given: The pair of linear equations: 1) \( x + y = 5 \) 2) \( 2x + 2y = 10 \) To Find: Whether the given pair of equations is consistent or inconsistent. If consistent, obtain the solution graphically. Formula: For two linear equations \( a_1x + b_1y = c_1 \) and \( a_2x + b_2y = c_2 \): If \( \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2} \), the equations are dependent and consistent (infinitely many solutions). If \( \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2} \), the equations are inconsistent (no solution). If \( \dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2} \), the equations are consistent and independent (unique solution). Graphic...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.1 Question 3(iii) Question: On comparing the ratios a_1 , b_1 , c_1 and a_2 , b_2 , c_2 , find out whether the following pair of linear equations are consistent or inconsistent: \[ \frac{3}{2}x + \frac{5}{3}y = 7; \] \[ 9x - 10y = 14. \] Given: The two linear equations are: (1) \( \frac{3}{2}x + \frac{5}{3}y = 7 \) (2) \( 9x - 10y = 14 \) To Find: Whether the given pair of linear equations is consistent or inconsistent by comparing the ratios: \[ \frac{a_1}{a_2}, \quad \frac{b_1}{b_2}, \quad \frac{c_1}{c_2} \] Formula: For a pair of linear equations: \[ a_1x + b_1y + c_1 = 0 \] \[ a_2x + b_2y + c_2 = 0 \] If \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \), the equations are consistent and have a unique solution. If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \), the equations are inconsistent (no solution). If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Question: Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: \( x - y = 8 \), \( 3x - 3y = 16 \) Given: The given pair of linear equations is: 1) \( x - y = 8 \) 2) \( 3x - 3y = 16 \) To Find: Whether the given pair of equations is consistent or inconsistent. If consistent, obtain the solution graphically. Formula: Two linear equations in two variables are consistent if they have at least one solution (i.e., they intersect or are coincident). For equations in the form \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \): Consistent if \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \), or \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \) (Inconsistent if all three ratios are equal). To solve graphically, plot both equations and find their point of intersection. Solution: Step 1: Write both eq...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.1 Question 4(iii) Question: Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: \( 2x + y - 6 = 0 \), \( 4x - 2y - 4 = 0 \) Given: Equation 1: \( 2x + y - 6 = 0 \) Equation 2: \( 4x - 2y - 4 = 0 \) To Find: Whether the given pair of equations is consistent or inconsistent. If consistent, find the solution graphically. Formula: For two linear equations in the form \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \): If \( \dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2} \), the system is consistent and has a unique solution. If \( \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2} \), the system is inconsistent (no solution). If \( \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2} \), the system has infinitely many solutions (consistent). Solution: Step 1: Write the equa...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Question: Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: \(2x - 2y - 2 = 0\), \(4x - 4y - 5 = 0\) Given: Pair of linear equations: 1) \(2x - 2y - 2 = 0\) 2) \(4x - 4y - 5 = 0\) To Find: Are the given pair of equations consistent or inconsistent? If consistent, obtain the solution graphically. Formula: General form of a linear equation: \(ax + by + c = 0\) To check consistency for equations \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\): If \(\dfrac{a_1}{a_2} \ne \dfrac{b_1}{b_2}\), the system is consistent and has a unique solution. If \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \ne \dfrac{c_1}{c_2}\), the system is inconsistent (no solution). If \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}\), the system is consistent with infinitely many solutions. Solution: Step 1: Write the equat...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Question: 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. Given: Let the width of the garden be \( x \) m. Length of the garden is \( x + 4 \) m. Half the perimeter is 36 m. To Find: The length and width of the garden. Formula: Perimeter of a rectangle \( = 2 \times (\text{Length} + \text{Width}) \) Half perimeter \( = \text{Length} + \text{Width} \) Solution: Step 1: Let the width of the garden be \( x \) m. Then, the length is \( x + 4 \) m. $$ \text{Width} = x \text{ m} \\ \text{Length} = x + 4 \text{ m} $$ Step 2: According to the question, half the perimeter is 36 m. So, $$ \text{Length} + \text{Width} = 36 \\ x + (x + 4) = 36 $$ Step 3: Simplify the equation. $$ x + x + 4 = 36 \\ 2x + 4 = 36 $$ Step 4: Subtract 4 from both sides. $$ 2x + 4 - 4 = 36 - 4 \\ 2x = 32 $$ ...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Question: 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 (ii) parallel lines (iii) coincident lines. Given: The first linear equation is: $$ 2x + 3y - 8 = 0 $$ To Find: Another linear equation in two variables such that the pair of equations represents: (i) Intersecting lines (ii) Parallel lines (iii) Coincident lines Formula: For two linear equations in the form: $$ \begin{aligned} a_1x + b_1y + c_1 &= 0 \\ a_2x + b_2y + c_2 &= 0 \end{aligned} $$ The nature of the pair is determined as follows: Intersecting lines: \(\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}\) Parallel lines: \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}\) Coincident lines: \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}\) Solution: St...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Question: 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. Given: Equation 1: \(x - y + 1 = 0\) Equation 2: \(3x + 2y - 12 = 0\) The x-axis (\(y = 0\)) To Find: The coordinates of the vertices of the triangle formed by the two lines and the x-axis. Graphical representation with the triangular region shaded. Formula: To find intersection of two lines: Solve the two equations simultaneously. To find intersection with x-axis: Substitute \(y = 0\) in the equation of the line. Solution: Step 1: Find the x-intercept of \(x - y + 1 = 0\). Substitute \(y = 0\): \[ x - 0 + 1 = 0 \implies x + 1 = 0 \implies x = -1 \] So, the intersection with the x-axis is \((-1, 0)\). Step 2: Find the x-intercept of \(3x + 2y - 12 =...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Question: On comparing the ratios \( a_1, b_1, c_1 \) and \( a_2, b_2, c_2 \), find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: \( 5x - 4y + 8 = 0 \) \( 7x + 6y - 9 = 0 \) Given: The pair of linear equations: \( 5x - 4y + 8 = 0 \) \( 7x + 6y - 9 = 0 \) To Find: Whether the lines represented by the given equations: Intersect at a point Are parallel Are coincident by comparing the ratios of their coefficients. Formula: For two linear equations in the form: \( a_1x + b_1y + c_1 = 0 \) \( a_2x + b_2y + c_2 = 0 \) Compare the ratios: \[ \frac{a_1}{a_2},\quad \frac{b_1}{b_2},\quad \frac{c_1}{c_2} \] Case 1: If \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \), the lines intersect at a point. Case 2: If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \), the lines are parallel. Case 3: If \( \frac{a_1...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.1 Question 2(iii) Question: On comparing the ratios a 1 , b 1 , c 1 and a 2 , b 2 , c 2 , find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: \(6x - 3y + 10 = 0\) \(2x - y + 9 = 0\) Given: The two linear equations are: 1) \(6x - 3y + 10 = 0\) 2) \(2x - y + 9 = 0\) To Find: Whether the lines represented by these equations: intersect at a point are parallel or are coincident by comparing the ratios \( \frac{a_1}{a_2}, \frac{b_1}{b_2}, \frac{c_1}{c_2} \). Formula: For a pair of linear equations in the form: \( a_1x + b_1y + c_1 = 0 \) \( a_2x + b_2y + c_2 = 0 \) The nature of the lines is determined by comparing the ratios: \( \frac{a_1}{a_2}, \frac{b_1}{b_2}, \frac{c_1}{c_2} \) Case 1: If \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \), the lines intersect at a unique point. Case 2: If \( \fr...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Question: On comparing the ratios \(a_1, b_1, c_1\) and \(a_2, b_2, c_2\), find out whether the following pair of linear equations are consistent or inconsistent. \(3x + 2y = 5\) ; \(2x - 3y = 7\). Given: The pair of linear equations is: \(3x + 2y = 5\) \(2x - 3y = 7\) To Find: Whether the given pair of linear equations are consistent or inconsistent by comparing the ratios of coefficients. Formula: For two linear equations in the form: \(a_1x + b_1y + c_1 = 0\) \(a_2x + b_2y + c_2 = 0\) The pair is: Consistent (unique solution) if \(\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}\) Inconsistent (no solution) if \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}\) Consistent (infinitely many solutions) if \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}\) Solution: Step 1: Write the equations in standard form. $$ \begin{align*} 3x + 2y &= 5 \implies 3x + 2y - 5...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Question: On comparing the ratios \( a_1, b_1, c_1 \) and \( a_2, b_2, c_2 \), find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: \( 9x + 3y + 12 = 0 \) \( 18x + 6y + 24 = 0 \) Given: The two linear equations are: \( 9x + 3y + 12 = 0 \) \( 18x + 6y + 24 = 0 \) To Find: Whether the lines represented by these equations: (a) intersect at a point, (b) are parallel, or (c) are coincident. Formula: For two linear equations: \( a_1x + b_1y + c_1 = 0 \) \( a_2x + b_2y + c_2 = 0 \) The nature of their graphs is determined by the ratios: Case 1: If \( \dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2} \), the lines intersect at a point (unique solution). Case 2: If \( \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2} \), the lines are parallel (no solution). Case 3: If \( \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfra...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Question: Form the pair of linear equations in the following problems, and find their solutions graphically. 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. Given: 5 pencils and 7 pens together cost ₹50. 7 pencils and 5 pens together cost ₹46. To Find: The cost of one pencil. The cost of one pen. Formula: Let the cost of one pencil be ₹\( x \). Let the cost of one pen be ₹\( y \). Linear equation in two variables: \( ax + by = c \). Graphical solution: The intersection point of the two lines represents the solution. Solution: Step 1: Let the cost of one pencil be ₹\( x \) and the cost of one pen be ₹\( y \). According to the question: 5 pencils and 7 pens: \( 5x + 7y = 50 \) 7 pencils and 5 pens: \( 7x + 5y = 46 \) So, the pair of linear equations is: ...