General Form of a Linear Equation in Two Variables

Q.

A shopkeeper gives books on rent for reading.

She takes a fixed charge for the first two days, and an additional charge for each day thereafter.

Latika paid $₹22$for a book kept for six days, while Anand paid $₹16$ for the book kept for four days.

Find the fixed charges and the charge for each extra day.

1. $₹13,₹2$

2. $₹10,₹3$

3. $₹12,₹9$

4. $₹14,₹5$

Q.

From Bangalore bus stand, if we buy 2 tickets to Malleshwaram and 3 tickets to Yeshwanthpur, the total cost is Rs. 46; but if we buy 3 tickets to Malleshwaram and 5 tickets to Yeshwanthpur, then the total cost is Rs. 74. Find the fares from Bangalore bus stand to Malleshwaram and Yeshwanthpur.

1. Rs. 10 and Rs. 8

2. Rs. 12 and Rs. 10

3. Rs. 10 and Rs. 12

4. Rs. 8 and Rs. 10

Q.

Find the value of $k$ for which the following system of linear equations has infinite solutions:

$$\begin{gathered} x+(k+1)y=5 \\ (k+1)x+9y=8k-1 \end{gathered}$$

1. $3$

2. $1$

3. $2$

4. $4$

Q.

If 2 is added to the numerator and denominator the fraction becomes $\frac{9}{10}$ and if 3 is subtracted from the numerator and denominator it become $\frac{4}{5}$. Find the fraction.

1. $\frac{7}{9}$.

2. $\frac{5}{8}$.

3. $\frac{1}{7}$.

4. $\frac{7}{8}$.

Q.

Determine the truth value of the statement $\mathrm{\forall }x\mathrm{\exists }y(xy=1)$ if the domain for the variables consists of?

1. The nonzero real numbers.

2. The nonzero integers.

3. The positive real numbers.

Q.

Rohit can row downstream 32 km in 4 hours and upstream 4km in 1 hour. Find his speed in still water and the speed of the current.

1. 6 km/hr, 2 km/hr

2. 4 km/hr, 4 km/hr

3. 4 km/hr, 2 km/hr

4. 20 km/hr, 4 km/hr

Q. A man can row 40 km upstream and 55 km downstream in 13 hours. Also, he can row 30 km upstream and 44 km downstream in 10 hours. Find the speed of the man in still water and the speed of the current.

1. 8 km/hr, 5 km/hr
2. 8 km/hr, 3 km/hr
3. 5 km/hr, 3 km/hr
4. 9 km/hr, 6 km/hr

Q. Express the linear equations. 2x-3y=5 in the standard form nd indicate the values of a, b, c

Q.

Select the correct approach for solving a pair of linear equations in 2 variables by elimination method.
i) Add or subtract one equation from the other so that one variable gets eliminated.
ii) Multiply both the equations by any non-zero constant to make the coefficients of one variable (either x or y) numerically equal.
iii) Solve the equation in one variable (x or y) to get its value.
iv) Substitute the value of x (or y) in either of the original equations to get the value of the other variable.

1. (ii), (i), (iii), (iv)

2. (i), (ii), (iii), (iv)

3. (i), (iv), (ii), (iii)

4. (ii), (i), (iv), (iii)

Q. Solve the following pair of linear equations:
$\frac{2}{\sqrt{x}}+\frac{3}{\sqrt{y}}=2$
$\frac{4}{\sqrt{x}}-\frac{9}{\sqrt{y}}=-1$
(where $x>0,y>0)$

1. $x=4,y=9$
2. $x=3,y=2$
3. $x=\frac{1}{2},y=\frac{1}{3}$
   
4. $x=9,y=4$

Q.

The length of a rectangle is twice the width of it. If the perimeter of the rectangle is 42m. Then the area of the rectangle is _____ .

1. 108 m²

2. 98 m²

3. 224 m²

4. 63 m²

Q.

If y = $\frac{1}{2}$ (3x+7) is written in the form ax + by + c = 0, then the values of b is -

1. -3

2. 2

3. 5

4. 7

Q. Question 1
The linear equation 2x - 5y = 7 has:

A) A unique solution
B) Two solution
C) Infinitely many solutions
D) No solution

Q. The graph of the linear equation 5x + 3y = 10 is a line that meets the x-axis at the point:

1. (0, 3)
   
   

2. (3, 0)
   
   

3. (2, 0)
   
   
4. (0, 2)
   
   

Q.

Compare the given linear equation $2y=5x+7$ with its standard form $ax+by+c=0.$
Which of the following is the possible value of 'c' for the given equation?

1. 7

2. 1/2

3. 2

4. -7/2

Q.

A man can row 40 km upstream and 55 km downstream in 13 hours. He can also row 30 km upstream and 44 km downstream in 10 hours. Find the speed of the boat in still water and the speed of the current.

1. 8 km/hr, 5 km/hr

2. 8 km/hr, 3 km/hr

3. 5 km/hr, 3 km/hr

4. 9 km/hr, 6 km/hr

Q.

Any equation which can be put in the form $ax+by+c=0$ where are $a,b,c$ are $\underline{\left(P\right)}$ and $a$ and $b$ are not both $\underline{\left(Q\right)}$ , is called linear equation in $\underline{\left(R\right)}$ .

1. $\left(P\right)$-constants, $\left(Q\right)$-zero , $\left(R\right)$-three variables

2. $\left(P\right)$-constants, $\left(Q\right)$-real number , $\left(R\right)$-three variables

3. $\left(P\right)$-real number, $\left(Q\right)$-zero , $\left(R\right)$-two variables

4. $\left(P\right)$-variables, $\left(Q\right)$-constants , $\left(R\right)$-two variables

Q.

Solve

3x + 2y = 36

5x + 4y = 64

1. -

2. 

3. 2

4. 7

Q.

If the linear equation y = 7x - 4 is written in standard form, the standard form being ax +by +c =0, then what is the value of a, b and c?

1. -7, -1, -4

2. none of these

3. 7, -1, 4

4. -7, 1, 4

Q.

The conditions for $ax+by+c=0$ to be a linear equation in two variables $(x,y)$ are

1. $a=0andb=0$

2. $a\ne 0andb\ne 0$

3. $a$ can be zero but not $b$

4. $b$ can be 0 but not $a$

Q.

If the linear equation y = 7x - 4 is written in the standard form ax +by +c =0, then what are the values of a, b and c?

1. -7, 1, 4

2. -7, -1, -4

3. 7, -1, 4

4. none of these

Q. Points A and B are 70 km apart on a highway. A car starts from A and another car starts from B at the same time. If they travel in the same direction, they meet in 7 hours, but if they travel towards each other they meet in one hour. What are their speeds in km/hr?

1. $40,20$
2. $40,30$
3. $50,30$
4. None of these

Q.

It takes 3 hours for a boat to travel 27 km upstream and the same boat can travel 30 km downstream in 2 hours. The speed of the current is 12 km/hr

1. True

2. False

Q.

The cost of tickets for a movie show was ₹162 for 12 children and 3 adults and was ₹ 122 for 8 children and 3 adults. The ticket cost for each child is ₹10 and each adult is ₹14.

1. False

2. True

Q.

If Nishu can swim downstream at $6km/h$ and upstream at $2km/h$. What is his speed in still water?

Q.

Solve the following pair of equations:
$\frac{1}{x}+\frac{3}{y}=1$
$\frac{6}{x}-\frac{12}{y}=2$
(Where $x\ne 0,y\ne 0)$

1. $x=\frac{5}{3},y=\frac{15}{2}$
2. $x=3,y=11$
3. $x=4,y=9$
4. $x=\frac{3}{5},y=\frac{7}{3}$

Q.

What will you substitute to make the following equation a linear equation?

$\frac{1}{x}$ + $\frac{3}{y}$ = 1

1. 1/x as p and 1/y as q

2. Option A or Option B

3. None of the above

4. x as 1/p and y as 1/q

Q. Question 1
Which one of the following options is true, and why?

$y=3x+5$ has
(i) a unique solution
(ii) only two solutions
(iii) infinitely many solutions

Q.

Points A and B are 90 km apart on a highway. A car starts from A and another from B at the same time. If they travel in the same direction they meet in 9 hours and if they travel in opposite directions they meet in $\frac{9}{7}$ hours. Speed of car starting from point A is:

1. 90 km/hr

2. 40 km/hr

3. 220 km/hr

4. 30 km/hr

Q.

A boat covers a distance of 32 km upstream and 36 km downstream in 7 hours. It also covers a distance of 40 km upstream and 48 km downstream in 9 hours. Find the speed of the boat in still water and that of the stream.

1. 44 km/hr, 8 km/hr

2. 14 km/hr, 5 km/hr

3. 6 km/hr, 4 km/hr

4. 10 km/hr, 2 km/hr