Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method

[question] Question. Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method (i) 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 Rs. 1000 as hostel charges whereas a student B, who takes food for 26 days, pays Rs. 1180 as hostel charges. Find the fixed charges and the cost of food per day. (ii) A frac...

Solve the following pair of linear equations by the substitution and cross-multiplication methods:

[question] Question. Solve the following pair of linear equations by the substitution and cross-multiplication methods: 8x + 5y = 9, 3x + 2y = 4 [/question] [solution] Solution: By substitution method, 8x + 5y = 9 ...(i) 3x + 2y = 4 ...(ii) From (ii), we get $x=\frac{4-2 y}{3}$ Substituting x from (iii) in (i), we get $8\left(\frac{4-2 y}{3}\right)+5 y=9$ = 32 – 16y + 15y = 27 = 5 = y Substituting y = 5 in (ii) we get 3x + 2(v) = 4 = 3x = – 6 = x = – 2 Hence, x = – 2, y = 5 By cross multiplicati...

(i) For which values of a and b does the following pair of linear equations have an infinite number of solutions?

[question] Question. (i) For which values of a and b does the following pair of linear equations have an infinite number of solutions? 2x + 3y = 7 (a – b) x + (a + b) y = 3 a + b – 2 (ii) For which value of k will the following pair of linear equations have no solution? 3x + y = 1 (2k – 1) x + (k –1) y = 2k + 1. [/question] [solution] Solution: (i) 2x + 3y – 7 = 0 ...(i) (a – b) x + (a + b) y – (3 a + b – 2) = 0 ...(ii) For infinite number of solutions, we have $\frac{\mathbf{a}-\mathbf{b}}{\mat...

Which of the following pairs of linear equations has unique solution, no solution, or infi nitely many solutions.

[question] Question. Which of the following pairs of linear equations has unique solution, no solution, or infi nitely many solutions. In case there is a unique solution, find it by using cross multiplication method. (i) x – 3y – 3 = 0 3x – 9y – 2 = 0 (ii) 2x + y = 5 3x + 2y = 8 (iii) 3x – 5y = 20 6x – 10y = 40 (iv) x – 3 y – 7 = 0 3x – 3y – 15 = 0 [/question] [solution] Solution: (i) x – 3y – 3 = 0, 3x – 9y – 2 = 0 $\frac{a_{1}}{a_{2}}=\frac{1}{3}, \frac{h_{1}}{b_{2}}=\frac{-3}{-9}=\frac{1}{3},...

Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method :

[question] Question. Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method : (i) If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1 . It becomes $\frac{\mathbf{1}}{\mathbf{2}}$ if we only add 1 to the denominator. What is the fraction? (ii) Five years ago Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu? (iii) The sum of the...

Solve the following pair of equations by the elimination method and the substitution method :

[question] Question. Solve the following pair of equations by the elimination method and the substitution method : (i) x + y = 5 and 2x – 3y = 4 (ii) 3x + 4y = 10 and 2x – 2y = 2 (iii) 3x – 5y – 4 = 0 and 9x = 2y + 7 (iv) $\frac{\mathbf{x}}{\mathbf{z}}+\frac{\mathbf{2} \mathbf{y}}{\mathbf{3}}=-\mathbf{1}$ and $x-\frac{\mathbf{y}}{\mathbf{3}}=3$ [/question] [solution] Solution: (i) Solution By Elimination Method: x + y = 5 ...(i) 2x – 3y = 4 ...(ii) Multiplying (i) by 3 and (ii) by 1 and adding w...

From the pair of linear equations for the following problems and find their solution by substitution method.

[question] Question. From the pair of linear equations for the following problems and find their solution by substitution method. (i) The difference between two numbers is 26 and one number is three times the other. Find them. (ii) The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them. (iii) The coach of a cricket team buys 7 bats and 6 balls for Rs. 3800. Later, she buys 3 bats and 5 balls for Rs. 1750. Find the cost of each bat and each ball. (iv) The taxi charges...

Solve 2x + 3y = 11 and 2x – 4y = – 24

[question] Question. Solve $2 x+3 y=11$ and $2 x-4 y=-24$ and hence tind the value of ' $m$ ' for which $y$ $=m x+3$ [/question] [solution] Solution: 2x + 3y = 11 ......(i) 2x – 4y = –24 ......(ii) Subtract equation (ii) from (i), we get 2x + 3y – 2x + 4y = 11 + 24 7y = 35 y = 5 Substituting value of y in equation (i), we get 2x + 3 × 5 = 11 2x = 11 – 15 $x=-\frac{4}{2}=-2$ Now, x = –2, y = 5 Puting value of x & y in y = mx + 3 5 = –2m + 3 2 = –2m m = –1 [/solution]...

Solve the following pair of linear equations by the substitution method.

[question] Question. Solve the following pair of linear equations by the substitution method. (i) x + y = 14, x – y = 4 (ii) $\mathrm{s}-\mathrm{t}=3, \frac{\mathbf{s}}{\mathbf{3}}+\frac{\mathbf{t}}{\mathbf{2}}=\mathbf{6}$ (iii) 3x – y = 3, 9x – 3y = 9 (iv) 0.2x + 0.3y = 1.3, 0.4x + 0.5y = 2.3 (v) $\sqrt{\mathbf{2}} \mathbf{x}+\sqrt{\mathbf{3}} \mathbf{y}=\mathbf{0}, \sqrt{\mathbf{3}} \mathbf{x}-\sqrt{\mathbf{8}} \mathbf{y}=\mathbf{0}$ (vi) $\frac{3 x}{2}-\frac{5 y}{3}=-2, \frac{x}{3}+\frac{y}{2...

Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0

[question] 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. [/question] [solution] Solution: x – y + 1 = 0 ...(i) 3x + 2y – 12 = 0 ...(ii) The vertices of the triangle are A (2, 3), B (–1, 0) and C (4, 0) [/solution]...

Given the linear equation 2x + 3y – 8 = 0

[question] Question. Given the linear equation $2 x+3 y-8=0$, 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 [/question] [solution] Solution: (i) 2x + 3y – 8 = 0 (Given equation) 3x + 2y + 4 = 0 (New equation) Here, $\frac{\mathbf{a}_{1}}{\mathbf{a}_{2}} \neq \frac{\mathbf{b}_{1}}{\mathbf{b}_{2}}$ Hence, the graph of the two equations will be two intersecting line...

Which of the following pairs of linear equations are consistent/inconsistent?

[question] Question. Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically : (i) $x+y=5.2 x+2 y=10$ (ii) $x-y=8,3 x-3 y=16$ (iii) $2 x+y-6=0,4 x-2 y-4=0$ (iv) $2 x-2 y-2=0,4 x-4 y-5=0$ [/question] [solution] Solution: (i) $x+y=5$ ...(i) $2 x+2 y=10 \quad \ldots$ ...(ii) $\frac{a_{1}}{a_{2}}=\frac{1}{2}, \frac{b_{1}}{b_{2}}=\frac{1}{2}, \frac{c_{1}}{c_{2}}=\frac{-5}{-10}=\frac{1}{2}$ i.c., $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{...

On comparing the ratios $\frac{\mathbf{a}_{1}}{\mathbf{a}_{2}}, \frac{\mathbf{b}_{1}}{\mathbf{b}_{2}}$ and $\frac{\mathbf{c}_{1}}{\mathbf{c}_{\mathbf{z}}}$

[question] Question. On comparing the ratios $\frac{\mathbf{a}_{1}}{\mathbf{a}_{2}}, \frac{\mathbf{b}_{1}}{\mathbf{b}_{2}}$ and $\frac{\mathbf{c}_{1}}{\mathbf{c}_{2}}$, find out whether the following pairs of linear equations are consistent, or inconsistent. (i) $3 x+2 y=5 ; 2 x-3 y=7$ (ii) $2 x-3 y=8 ; 4 x-6 y=9$ (iii) $\frac{\mathbf{3}}{\mathbf{z}} \mathrm{x}+\frac{\mathbf{5}}{\mathbf{3}} \mathrm{y}=7 ; 9 \mathrm{x}-10 \mathrm{y}=14$ (iv) $5 x-3 y=11 ;-10 x+6 y=-22$ (v) $\frac{\mathbf{4}}{\mat...

On comparing the ratios $\frac{\mathbf{a}_{1}}{\mathbf{a}_{8}}, \frac{\mathbf{b}_{1}}{\mathbf{b}_{2}}$ and $\frac{\mathbf{c}_{1}}{\mathbf{c}_{2}}$, find out whether the lines representing the

[question] Question. On comparing the ratios $\frac{\mathbf{a}_{1}}{\mathbf{a}_{2}}, \frac{\mathbf{b}_{1}}{\mathbf{b}_{2}}$ and $\frac{\mathbf{c}_{1}}{\mathbf{c}_{2}}$, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident. (i) $5 x-4 y+8=0 ; 7 x+6 y-9=0$ (ii) $9 x+3 y+12=0 ; 18 x+6 y+24=0$ (iii) $6 x-3 y+10=0 ; 2 x-y+9=0$ [/question] [solution] Solution: (i) $5 x-4 y+8=0$ ...(i) $7 x+6 y-9=0$ ...(ii) $\frac{a_{1}}{a_{2}}...

Form the pair of linear equations in the following problems, and find their solutions graphi cally

[question] Question. Form the pair of linear equations in the following problems, and find their solutions graphi cally (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. (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. [solution] Solution: (i) Let the number of boys be x and th...

The cost of 2 kg of apples and 1 kg of grapes on a day was found to be ` 160.

[question] Question. The cost of 2 kg of apples and 1 kg of grapes on a day was found to be ` 160. After a month, the cost of 4 kg of apples and 2 kg of grapes is ` 300. Represent the situation algebraically and geometrically. [/question] [solution] Solution: Let the cost of 1 kg of apple be ` x and the cost of 1 kg of grapes be ` y So, $2 x+y=160$ $4 x+2 y=300 x$ and [/solution]...

The coach of a cricket team buys 3 bats and 6 balls for ` 3900.

[question] Question. The coach of a cricket team buys 3 bats and 6 balls for ` 3900. Later, she buys another bat and 3 more balls of the same kind for ` 1300. Represent this situation algebraically and geometrically [/question] [solution] Solution: Let the cost of 1 bat be ` x and the cost of 1 ball be ` y So, $3 x+6 y=3900$ and $x+3 y=1300$ and [/solution]...

Aftab tells his daughter, "Seven years ago, I was seven times as old as you were then

[question] Question. Aftab tells his daughter, "Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be". (Isn't this interesting?) Represent this situation algebraically and graphically. [/question] [solution] Solution: Let the present age of Aftab's daughter = x years. and the present age of $A f t a b=y$ years $(yx)$ According to the given conditions Seven years ago, $(y-7)=7 \times(x-7)$ i.e., $\quad y-7=7 x-49$ i.e...

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