Finding Solution for Consistent Pair of Linear Equations

Q.

Solve for $x:\frac{1}{2a+b+2x}=\frac{1}{2a}+\frac{1}{b}+\frac{1}{2x}$

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Q.

Place A and B are 160 km apart on a highway.One car starts from A and another from B at the same time.If they travel in the same direction, they meet in 8 hours.But, If they travel towards each other, they meet in 2 hours.Find the speed of each car.

Q.

A man has some hens and cows.If the number of heads be 48 and number of feet be 140, how many cows are there?

Q. If in a rectangle, the length is increased and breadth reduced each by 2 units, the area is reduced by 28 square units. If, however the length is reduced by 1 unit and the breadth increased by 2 units, the area increases by 33 square units. Find the area of the rectangle.

Q. Solve the following simultaneous linear equations using the elimination method.

3x + 2y = 29
5x - y = 18

1. x = 7; y = 5
2. x = 5; y = 7
3. x = 8; y = 10
4. x = 10; y = 8

Q. In a number of two digits, unit's digit is twice the tens digit. If 36 be added to the number, the digits are reversed. The number is

1. 63
2. 84
3. 36
4. 48

Q. 3/x -1/y = -9 , 2/x +3/y = 5 . By substituting method

Q.

Solve the following pair of linear equations by elimination method : 3x + 2y = 8 and 2x – y = 3. [2 MARKS]

Q.

Represent the following situations in the form of quadratic equations.

(i) The area of a rectangular plot is 528 m². The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.

(ii) The product of two consecutive positive integers is 306. We need to find the integers.

(iii) Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age.

(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.

Q.

ABCD is a cyclic quadrilateral. Find the angles of the cyclic quadrilateral.

Q.

Find the value of x?

x + y = 5
3x - y = 3

1. 3

2. -3

3. -2

4. 2

Q.

If the area of a rectangle is 24 m² and its perimeter is 20 m, the equation to find its length and breadth would be:

1. x² + 10x + 28 = 0

2. x² - 10x - 24 = 0

3. x² - 10x + 24 = 0

4. x² + 1 2x + 24 = 0

Q. A man starts his job with a certain salary and earns a fixed increment every year. If his salary was ₹15000 after 4 years of service and ₹18000 after 10 years of service, then find his starting salary and annual increment respectively.

1. ₹11000, ₹700
2. ₹13000, ₹500
3. ₹13000, ₹700
4. ₹11000, ₹500

Q.

Two cars leave simultaneously from points A and B, the distance between which is $280km$.

If the cars move to meet each other, they’ll meet in $2$ hours.

But if they move in the same direction, then the car going from point $A$ will catch up with the car going from point $B$ in $14$ hours.

What is the speed of each of the cars?

Q.

Let y(x) be a solution of the differential equation $(1+e^x)y^{\prime }+y{e}^x=1.Ify\left(0\right)=2,$ then which of the following statement (s) is/are true ?

1. y (-4) = 1

2. y(-2) = 0

3. y(x) has a critical point in the interval (-1, 0)

4. y(x) has no critical point in the interval (-1, 0)

Q.

Statement 1 : The graph of a linear equation is a straight line.

Statement 2 : The solution of a pair of linear equations is a point common to two lines.

1. S1 is true but S2 is false

2. S1 is false but S2 is true

3. S1 and S2 are true

4. S1 and S2 are false

Q.

The length of the plot in meters is 1 more than twice its breadth and the area of a rectangle plot is 528 m². Which of the following quadratic equations represents the given situation:

1. 2x²+x+ 528=0

2. 2x²+x- 528=0

3. x²+2x- 528=0

4. x²+x- 528=0

Q.

Each angle of an equilateral triangle is equal to

Q. Question 4 (iv)
Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method:
Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

Q. Question 2 (i)
Formulate the following problems as a pair of equations, and hence find their solutions:
Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current.

Q.

Kims age is twice that of her sister. When you add Kims age to her sisters age, you get $36$. How old is each sister? (a) Write an equation that represents the situation. Explain any variable used. (b) Solve the equation from Part (a). Show your work. State your solution as a complete sentence.

Q.

148x+231y=910

231x+ 148y=610

find x and y

Q. In a $△ABC;\angle C=3\angle B=2(\angle A+\angle B)$. Find the three angles.

Q. Solve the following pair of equations using cross multiplication method.
x + 2y = -2 , x - y = 4

Q. Points A and B are $70km$ apart on a highway. A car starts from A and another starts from B simultaneously. If they travel in the same direction, they meet in 7 hours. But, if they travel towards each other, they meet in 1 hour. Find the speed of each car in km/hr.​

1. 40 and 30
2. 40 and 45
3. 30 and 25
4. 30 and 20

Q.

Form the equation.

Two-digit number, whose tens digit is $\mathrm{x}$ and units digit is $\mathrm{y}$.

Q.

The first team of workers received $50kg$ of cement less than the second team. For every hour of work, the first team was using $150kg$ of cement, while the second team was using $200kg$.In three hours the first team had $1.5$ times as much remaining cement as the second team. How much cement was delivered to each team of workers?

Q. Question 3 (i)
Form 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.

Q.

The sum of the ages of a man and his son is $45$ years. Five years ago, the product of their age was four times the mans age at that time. Find their present ages.

Q. Question 3 (vi)
Form the pair of linear equations for the following problems and find their solution by substitution method:
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?