Pair of Linear Equations in Two Variables

What is the intersection point of the line $x + y = 0$ and the x-axis?

The value of $c$ for which the pair of equations $cx - y = 2$ and $6x - 2y = 3$ will have infinitely many solutions is:

Graphically, the pair of equations $6x - 3y + 10 = 0$ and $2x - y + 9 = 0$ represents two lines which are:

If a pair of linear equations is consistent, then the lines will be:

The pair of equations $x + 2y = 5$ and $3x + 6y = 15$ has how many solutions?

Which of the following points lies on the line $y - x = 2$ and is also 5 units away from the origin on the same line in the first quadrant?

The lines $x = a$ and $y = b$ are graphically:

Two drones, A and B, follow linear paths given by $x - y = 1$ and $2x + 3y = 12$. A technician plots these paths to ensure they don't collide at an unexpected height. Looking at the graph, the technician also identifies the region bounded by these two lines and the x-axis. Find the vertices of the triangle formed by these lines and the x-axis.

An architect is designing a rectangular park. The perimeter is 32 meters. If the length ($x$) is increased by 2 meters and the breadth ($y$) is decreased by 2 meters, the area remains the same. By plotting the system of equations representing the perimeter and the change in dimensions, determine the original dimensions of the park.

A logistics company manages two delivery routes modeled by the equations $3x - 2y = 4$ and $x + y = 3$. The operations manager plots these on a coordinate plane where $x$ represents the distance from the hub (in km) and $y$ represents the fuel consumption (in liters). At what coordinate $(x, y)$ do the two routes intersect, indicating a point of identical efficiency, and what is the area of the triangle formed by these two lines and the y-axis?

For what value of $k$ will the system of linear equations $x + 2y = 3$ and $5x + ky + 7 = 0$ represent a pair of parallel lines on a coordinate plane?

Which algebraic condition matches the visual representation of the consistent and independent system shown?

If the lines $3x + 2ky = 2$ and $2x + 5y + 1 = 0$ are parallel, as indicated in the diagram (slopes are equal), what is the value of $k$?

Two lines $L_1$ and $L_2$ are shown. $L_1$ is $2x + 3y = 5$. If $L_2$ is $4x + 6y = 10$, what is the relationship between the lines?

Two linear equations are represented graphically by the lines $L_1$ and $L_2$ as shown in the coordinate plane. Based on the intersection point and the algebraic conditions for a pair of linear equations $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$, which of the following describes the system?

A student is analyzing a system of two linear equations. The algebraic condition is found to be $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$. If the first line is $2x + 3y = 6$, which of the following could be the second line such that the system is inconsistent?

Consider the system of equations $ax + by = c$ and $lx + my = n$. If the visual representation shows the lines intersecting at the origin $(0,0)$ and the ratio $\frac{a}{l} \neq \frac{b}{m}$, which of the following must be true about the constants $c$ and $n$?

A logistics firm manages two delivery routes, A and B. The distance covered on Route A is represented by the equation $2x + (k-1)y = 4$ and Route B by $(k+1)x + 3y = k+6$, where $x$ and $y$ represent coordination parameters. An analyst determines that these routes are represented by two parallel lines on the supply chain map, meaning they never intersect and the system is inconsistent. Based on this information, find the specific value of $k$ that satisfies this condition.

A telecommunications provider uses two signal towers. The coverage area boundaries are defined by the linear equations $x + 2y = 5$ and $(a-b)x + (a+b)y = 15$. For the signals to perfectly coincide (infinitely many solutions/consistent dependent), ensuring no dead zones, what must be the values of $a$ and $b$?

A logistics company maps its delivery routes on a coordinate plane. Route A is represented by the equation $2x + 3y = 12$. The company wants to add Route B such that the system of paths is inconsistent (parallel), meaning the delivery vans never meet. If Route B passes through the point $(0, 6)$, what is the value of the constant $k$ in the equation $4x + 6y = k$?

Solve the system $x - 2y = 0$ and $x + y = 6$.

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

Solve the pair $x + y = 11$ and $x - y = 3$ using elimination.

Solve using substitution: $x + 3y = 7$ and $x = 1$.

Find the point of intersection of the lines represented by $3x - 2y = 6$ and $x + y = 7$.

Solve the equations: $4x + 0.5y = 4.5$ and $3x - 0.4y = 2.6$.

Solve for $x$ and $y$: $\frac{x+y-8}{2} = \frac{x+2y-14}{3} = \frac{3x+y-12}{11}$.

Solve for $x$ and $y$: $x/a^2 + y/b^2 = 2$ and $x/a - y/b = 0$.

Find $x$ and $y$: $\frac{xy}{x+y} = \frac{1}{a}$ and $\frac{xy}{x-y} = \frac{1}{b}$.

Solve for $x$: $\sqrt{a}x - \sqrt{b}y = 0$ and $\sqrt{ab^2}x + \sqrt{a^2b}y = a+b$.

The difference between two numbers is 5 and the difference between their squares is 65. Find the larger number.

An algebraic expression for 'the sum of two numbers $x$ and $y$ is 15' and 'their difference is 5' is given by which pair of equations?

A shopkeeper sells a saree for $\text{₹} 800$ and a sweater for $\text{₹} 400$, earning a profit of $\text{₹} 100$. If he sells the saree for $\text{₹} 600$ and the sweater at its cost price, he has no profit or loss. What is the cost price of the saree?

The age of a woman is twice the age of her daughter. Ten years ago, the woman was three times as old as her daughter. Find the daughter's current age.

In a factory, the daily wages for $6$ supervisors and $8$ laborers amount to $₹5,600$. If there are $4$ supervisors and $10$ laborers, the total daily wages amount to $₹4,900$. Find the daily wage of one supervisor.

The sum of two numbers is 35 and their difference is 13. Find the numbers.

A father is 3 times as old as his son. In 12 years, he will be twice as old as his son. Find the present age of the son.

If the length of a rectangle is reduced by $5$ units and its breadth is increased by $2$ units, its area is reduced by $80$ sq units. However, if we increase the length by $10$ units and decrease the breadth by $5$ units, the area is increased by $50$ sq units. Find the length and breadth of the rectangle.

A total of $Rs.\ 50000$ is invested in two bonds. The first bond yields an interest of $8\%$ per annum and the second bond yields $10\%$ per annum. If the total annual interest is $Rs.\ 4400$, how much was invested in each bond?

The sum of a two-digit number and the number obtained by interchanging the order of its digits is $165$. If the digits differ by $3$, find the number (where the tens digit is larger).

The point $(0, 4)$ lies on the graph of which equation?

A linear equation in two variables $x$ and $y$ is of the form $ax + by + c = 0$, where $a, b, c$ are:

Which of the following values of $(x, y)$ satisfies the equation $x - 2y = 0$?

The pair of linear equations $2x + 3y = 5$ and $4x + 6y = 15$ has:

Calculate the area of the quadrilateral region bounded by the lines $x = 2$, $x = 5$, $y = 1$, and $y = 4$.

Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. If $x$ and $y$ are the current ages of Nuri and Sonu respectively, calculate the value of $x - y$.

If the line $ax - y = 5$ passes through the point of intersection of the lines $x - 2y = 1$ and $x - y = 2$, find the value of $a$.

A triangle is formed by the coordinate axes and the line $3x + 2y = 12$. If a horizontal line $y = k$ divides the area of this triangle into two equal parts, what is the value of $k$?

The value of $k$ for which the system $x + (k+1)y = 5$ and $(k+1)x + 9y = 8k-1$ has infinitely many solutions is:

Find the value of $k$ for which the system $x + 2y = 3$ and $5x + ky + 7 = 0$ is inconsistent.

Which condition for the system $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ indicates a unique solution?

The graph of the equation $2x + 3y = 6$ cuts the x-axis at the point:

In the graph of a system of linear equations, if the lines are parallel, how many solutions exist?

The pair of equations $x = a$ and $y = b$ graphically represents lines which are:

The graph of a pair of linear equations $x - y = 1$ and $2x + y = 8$ is shown in the coordinate plane. Find the coordinates of the point of intersection of these two lines, which represents the solution to the pair of equations.

The graph shows two parallel lines representing the equations $x - 2y = 2$ and $2x - 4y = 12$. Based on the graphical method, how many solutions does this pair of linear equations have?

Two lines $L_1$ and $L_2$ are plotted on the coordinate plane. $L_1$ passes through $(0, -2)$ and $(2, 0)$, while $L_2$ passes through $(0, 4)$ and $(4, 0)$. Based on the graphical representation, what is the solution $(x, y)$ for this pair of equations?

A tech company is analyzing the cost of two different cloud storage plans. Plan A has a fixed monthly maintenance fee of Rs 200 plus Rs 100 per terabyte of data used. Plan B has no maintenance fee but costs Rs 200 per terabyte. The graph shows the linear equations representing the total monthly cost ($y$) for $x$ terabytes of data. Based on the graphical intersection, at what amount of data usage will both plans cost the same, and what is that cost?

The path of two boats A and B are given by $x - y = 1$ and $2x + 3y = 12$. The graph shows their paths. A circular buoy is placed at the intersection of these paths. If a third boat C follows the path $y = 0$ (the x-axis), find the coordinates of the vertices of the triangle formed by the paths of Boat A, Boat B, and Boat C.

Two parallel tracks are represented by the equations $x + 2y - 4 = 0$ and $2x + 4y - 12 = 0$. A maintenance drone is positioned at $(0,0)$. Based on the graphical representation of these lines, what can be concluded about the possibility of these tracks crossing each other at any point in the coordinate plane?

Determine the value of $x$ for the system $x + y = 5$ and $2x + 2y = 10$.

Solve for $y$: $2x + 3y = 7$ and $2x + y = 5$.

If $x - y = 4$ and $x + y = 10$, find the value of $y$.

Solve the pair of linear equations $x + y = 7$ and $3x - y = 1$ for $x$.

The cost of 5 oranges and 3 apples is $Rs. 35$ and the cost of 2 oranges and 4 apples is $Rs. 28$. Find the cost of one orange.

Find $x$ and $y$: $\frac{5}{x - 1} + \frac{1}{y - 2} = 2$ and $\frac{6}{x - 1} - \frac{3}{y - 2} = 1$.

Solve for $x$ and $y$: $px + qy = p - q$ and $qx - py = p + q$.

For what value of $k$ will the equations $x+2y+7=0$ and $2x+ky+14=0$ represent coincident lines?

Solve for $x$ given $ax+by=a-b$ and $bx-ay=a+b$.

Given $x+y=5$ and $2x-3y=4$, find the value of $x^2-y^2$.