Linear Equations

The difference between a mother's age and the sum of her two daughter's age is $8$. The average age of two daughters is $24$. What is the age of the mother?

If $5^{5x+5}= 1,$ then x is equal to -

If value of $x=1+\sqrt{2}+\sqrt{3}$, then value of $\left(\frac{x+1}{x-1}\right)=?$

Study the following information and answer the question given below: Twelve friends $A, B, C, D, E, F, G, H, I, J, K$and $L$ were born in different months of the same year. $A$ was born in the month of April and $G$ was born in the month of August. $J$ was born the month immediately preceding the month in which $K$ was born and immediately succeeding the month in which $C$ was born. $J$ was not born in the month of October nor in February. There is a gap of two months between the birthdays of $L$ and $B$. There were $30 \mathrm{days}$ in the month in which $L$ was born. $D$ was born in the month immediately after the month in which $I$ was born. There were $31 \mathrm{days}$ in the month in which $D$ was born. There is a gap of one month between the birthdays of $B, F, E$ and $H$ were born in that months which had $31 \mathrm{days}$ each.

Who among the following was born in the month of February ?

In the question, two equations numbered I and II with variables $x$ and $y$ are given. You have to solve both the equations to find the value of $x$ and $y$. Give answer:
A. If $x>y$
B. If $x\ge y$
C. If $x<y$
D. If $x\le y$
E. If $x=y$ or relationship between $x\text{ and} y$cannot be determined.

I. $4x^2-4 = 60$
II. $3y^2+3= 51$

Study the following information and answer the question given below: Twelve friends $A, B, C, D, E, F, G, H, I, J, K$ and $L$ were born in different months of the same year. $A$ was born in the month of April and $G$ was born in the month of August. $J$ was born the month immediately preceding the month in which $K$ was born and immediately succeeding the month in which $C$ was born. $J$ was not born in the month of October nor in February. There is a gap of two months between the birthdays of $L$ and $B$. There were $30$ days in the month in which $L$ was born. $D$ was born in the month immediately after the month in which $I$ was born. There were $31$ days in the month in which $D$ was born. There is a gap of one month between the birthdays of $B, F, E$ and $H$ were born in that months which had $31$ days each.

Four of the following five are alike in a certain way and hence form a group. Which one of the following does not belong to that group?

The sum of the present age of $P, Q$ and $R$ is $X$. Present age of $R$ is $X – 50$. After $12 \mathrm{years}$, the age of $R$ is equal to the present age of $Q$. After $6 \mathrm{years}$, the age of $P$ is equal to the present age of $Q$. Find the present age of $R$.

Saravanan went shopping and purchase Pen and Pencil. When he observed that the price of $14$ pens is equal to $20$ Pencils he realised that he can purchase $500$ Pencils from the amount of money which he had but he further spends $10\%$ of his money on the packing charges. If he purchased $70$ pens and $290$ pencils, then how many pencils can he still purchase?

The difference of a three-digit number and the number formed by reversing its digits is more than $198$. The sum of the units and tens digit is the same as the difference of the hundredth and the unit digits and also the hundredth digit is twice of the unit digit. Find the number.

$$\begin{gathered} \mathrm{I}. 613 \\ \mathrm{II}. 603 \\ \mathrm{III}. 623 \\ \mathrm{IV}. 804 \\ \mathrm{V}. 834 \\ \mathrm{VI}. 824 \end{gathered}$$

Directions: In each of these questions, two equations (I) and (II) are given. You have to solve both the equations and give the answer.
$$\begin{gathered} \left(i\right)9x-8y=31 \\ \left(ii\right)18x+3y=33 \end{gathered}$$

Directions: In each of these questions, two equations (I) and (II) are given. You have to solve both the equations and give the answer.
$$\begin{gathered} \left(I\right).6x+8y=17 \\ \left(II\right).3y+2x=19 \end{gathered}$$

The distance between two stations is $300 \mathrm{km}$. Two trains start simultaneously from these stations and move towards each other. The speed of one of them is $7 \mathrm{km}/\mathrm{h}$ more than that of the other. If the distance between them after $2$ hours of their start is $34 \mathrm{km}$, find the speed of each train. 

In the question, two equations I and II with variables a and b are given. You have to solve both the equations to find the values of a and b.
I. $2a + 3b = 4$
II. $8a – 3b = \frac{7}{2}$

Three small and two large boxes can hold 43 packs of chocolate. One large box minus two small boxes constitute 18 packs of chocolate. How many packs of chocolate can each box hold?

What should come in place of question mark (?) in the following questions?

If $2x + 3y = 34$ and $\frac{x+y}{y}=\frac{13}{8}$  then $5y + 7x = ?$

What will come in place of the question mark (?) in the following question? (You are not expected to calculate exact values)$\frac{1576}{7}+124.98\times \frac{1}{5}=?-\frac{2457}{11}$

In each of these questions, two equations $\left(I\right)$ and $\left(II\right)$ are given. You have to solve both the equations and give answer.

$$\begin{gathered} I. 7m – 3n = 13 \\ II. 5m + 4n = 40 \end{gathered}$$

In each of these questions, two equations (I) and (II) are given. You have to solve both the equations and give the answer.

$$\begin{gathered} I. 5x² - 31x + 44 = 0 \\ II. 5y² +4y -33=0 \end{gathered}$$

In how many different ways can the letters of the word "EPENTHESIS" be arranged. so that vowels always come together?

If $3^{x-y}=27$ and $3^{x+y}=243$, then find the value of $x$.