Pair of Linear Equations in Two Variables

If a pair of linear equations $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ has a unique solution, then which of the following conditions is correct?

If the pair of linear equations $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ represents coincident lines, then:

The pair of equations $x + y = 14$ and $x - y = 4$ has the solution:

For what value of $k$ will the equations $3x - y - 5 = 0$ and $6x - 2y - k = 0$ have no solution?

The pair of equations $y = 0$ and $y = -7$ has:

If $2x + 3y = 11$ and $2x - 4y = -24$, then the value of $m$ in $y = mx + 3$ is:

If $2x + y = 7$ and $x + 2y = 8$, then the value of $x + y$ is:

If a pair of linear equations is consistent, then the lines represented by them are:

For what value of $k$ will the pair of equations $kx + 2y = 5$ and $3x + y = 1$ have a unique solution?

The pair of equations $x + 2y + 5 = 0$ and $-3x - 6y + 1 = 0$ has:

If $x = a, y = b$ is the solution of the equations $x - y = 2$ and $x + y = 4$, then the values of $a$ and $b$ are respectively:

For what value of $c$ does the pair of equations $cx - y = 2$ and $6x - 2y = 3$ have no solution?

If a pair of linear equations is inconsistent, then the graph of these equations will be:

For what value of $k$ does the pair of linear equations $x + 2y - 3 = 0$ and $5x + ky + 7 = 0$ represent parallel lines?

For what value of $k$ will the following pair of linear equations have infinitely many solutions? $$kx + 3y = k - 3$$ $$12x + ky = k$$

Define a pair of linear equations in two variables and state its standard general form.

Write the condition for a pair of linear equations in two variables to have a unique solution. Also explain its graphical interpretation.

Write the condition under which a pair of linear equations has no solution. What type of lines do they represent graphically?

Check whether the pair of linear equations $x + 2y - 4 = 0$ and $2x + 4y - 12 = 0$ is consistent or inconsistent.

Find the value of $k$ for which the system of equations $x + 2y = 3$ and $5x + ky + 7 = 0$ has a unique solution.

For what value of $k$ will the equations $2x + 3y = 7$ and $(k-1)x + (k+2)y = 3k$ have infinitely many solutions?

Solve the pair of linear equations $x + y = 14$ and $x - y = 4$ using the substitution method.

Solve the system of equations $2x + y = 5$ and $3x + 2y = 8$ using the elimination method.

The sum of two numbers is 35 and their difference is 13. Formulate a pair of linear equations and find the numbers.

If $x = a$ and $y = b$ is the solution of the linear equations $x - y = 2$ and $x + y = 4$, then find the values of $a$ and $b$.

Express $y$ in terms of $x$ from the linear equation $3x - 2y + 6 = 0$. Also, find the coordinates of the point where this line cuts the y-axis.

Explain the conditions for a pair of linear equations $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ to have: (i) a unique solution, (ii) infinitely many solutions, and (iii) no solution. What do these conditions represent graphically?

Solve the pair of linear equations $2x + 3y = 11$ and $2x - 4y = -24$ by substitution method. Hence, find the value of '$m$' for which $y = mx + 3$.

For what value of $k$ will the following pair of linear equations have infinitely many solutions? $$2x + 3y = 7$$ $$(k - 1)x + (k + 2)y = 3k$$

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 now?

A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid ₹27 for a book kept for seven days, while Susy paid ₹21 for the book she kept for five days. Find the fixed charge and the charge for each extra day.

Explain the conditions for a pair of linear equations in two variables to have a unique solution, infinitely many solutions, or no solution. Also, describe their geometric representation in each case.

Solve the following pair of linear equations by the elimination method: $$2x + 3y = 11$$ $$2x - 4y = -24$$\nHence, find the value of '$m$' for which $y = mx + 3$.

Explain the conditions for the consistency and inconsistency of a pair of linear equations in two variables: $$a_1x + b_1y + c_1 = 0$$ $$a_2x + b_2y + c_2 = 0$$\nDiscuss all three cases based on the ratios of their coefficients algebraically and graphically, providing a suitable example for each case.

Ritu can row downstream $20\text{ km}$ in $2\text{ hours}$, and upstream $4\text{ km}$ in $2\text{ hours}$.

1. Formulate the system of linear equations representing this situation.
2. Find her speed of rowing in still water and the speed of the current using the elimination method.
3. Show complete step-by-step calculations and verification.

Explain in detail the algebraic conditions for the consistency and inconsistency of a pair of linear equations in two variables. Discuss the graphical representation, geometric behavior, and the number of solutions for each condition using standard general equations.

A boat goes $30\text{ km}$ upstream and $44\text{ km}$ downstream in $10\text{ hours}$. In $13\text{ hours}$, it can go $40\text{ km}$ upstream and $55\text{ km}$ downstream. Formulate the pair of linear equations and find the speed of the stream and that of the boat in still water.