Linear Equations in Two Variables

A point $(2, 0)$ satisfies which of the following equations?

In the linear equation $y = mx$, if $x = 1$ and $y = 5$, then $m$ is:

If we divide both sides of the equation $4x + 6y = 12$ by 2, the new equation is:

The equation $x - 4 = \sqrt{3}y$ can be written in the form $ax + by + c = 0$ as:

Which of the following points is a solution to $x - 2y = 4$?

What are the values of $a, b,$ and $c$ if $x = -4$ is expressed in the form $ax + by + c = 0$?

If $2x + 5y = 10$, then what is the value of $x$ when $y = 2$?

A straight line is defined by $y - 3 = 2(x - 1)$. What is the y-intercept of this line?

If the point $(k, k^2)$ lies on the graph of $y = x + 6$, then the distance between the two possible points is:

The linear equation $y = 3x$ when represented in the form $ax + by + c = 0$ gives $a = 3, b = -1, c = 0$. If we scale the coefficients such that $a^2 + b^2 = 10$, what is the new value of $c$?

The equation $y = mx$ represents a line passing through:

The graph of the equation $x = y$ is a line that passes through which quadrant(s)?

Every point on the x-axis is of the form:

Which quadrant does NOT contain any point of the line $x + y = 2$?

The graph of the linear equation $2x + 3y = 6$ is drawn on the Cartesian plane. Find the coordinates of the point where the graph intersects the x-axis.

A line passes through the origin $(0,0)$ and the point $(2, 4)$ as shown in the graph. Which of the following linear equations represents this line?

What is the equation of the x-axis?

A force $F$ is applied to a spring, resulting in an extension $x$. The relationship is $F = kx$. If the graph passes through the point $(0.2, 10)$, where $x$ is in meters and $F$ is in Newtons, what is the value of $F$ when the extension $x$ is $0.5$ meters?

If the graph of the linear equation $3x + ky = 9$ passes through the point where the lines $x=1$ and $y=2$ intersect, find the value of $k$.

A linear equation $ax + by + c = 0$ passes through the points $(1, 4)$ and $(0, 6)$. What are the values of $a$, $b$, and $c$ such that the equation is simplified with $b=1$?

The graph of $x = k$ is a straight line:

If a point lies on the line $y = 10$, its distance from the x-axis is:

The line $x = -1$ passes through which quadrant(s)?

If a point $(3, a)$ lies on the x-axis, then the value of $a$ is:

The geometric representation of $x = -2$ as an equation in two variables is:

If a line is parallel to the x-axis and is 5 units below it, its equation is:

What is the coordinate of the point where the line $y = 3$ intersects the y-axis?

A line parallel to the $x$-axis passes through the point $(10, -8)$. What is the equation of the line that is the reflection of this line across the $x$-axis?

The distance between the line $x = a$ and the $y$-axis is twice the distance between the line $x = a$ and the line $x = 9$. If $0 < a < 9$, find the equation of the line.

If the line $y = k$ passes through the point $(5, 7)$, find the equation of a line parallel to the $x$-axis which is 10 units below $y = k$.

The graph of every linear equation in two variables is a:

In the equation $ax + by + c = 0$, $a, b,$ and $c$ are:

The solution of the pair of equations $x + y = 5$ and $x - y = 1$ is:

If $y$ is directly proportional to $x$, and $y=10$ when $x=2$, the equation is:

A boat covers $32$ km upstream and $36$ km downstream in $7$ hours. Also, it covers $40$ km upstream and $48$ km downstream in $9$ hours. Find the speed of the stream.

Two years ago, Dilip was thrice as old as his son and two years hence, twice his age will be equal to five times that of his son. Find Dilip's present age.

An isosceles triangle has a perimeter of $30$ cm and each of the equal sides is $12$ cm. Find the length of the third side.

The cost of $4$ tables and $3$ chairs is $Rs.\ 2250$ and that of $3$ tables and $4$ chairs is $Rs.\ 1950$. Find the cost of $1$ table.

A man started his job with a certain monthly salary and a fixed increment every year. If his salary was $Rs.\ 1500$ after $4$ years of service and $Rs.\ 1800$ after $10$ years of service, what was his starting salary?

If $x$ is the price of an apple and $y$ is the price of an orange, such that $5x + 3y = 35$ and $2x + 4y = 28$, find the cost of $3$ apples and $2$ oranges.

The graph of the equation $x + y = 4$ is shown. If another line $x - y = 0$ is drawn on the same plane, at which point will they intersect?

A pair of linear equations is represented by two lines $l_1$ and $l_2$ on a coordinate plane. If the lines intersect at the point $(2, 3)$, what is the solution to the system of equations?

In the graph of $y = 2$, what is the y-coordinate for any value of x?

What is the area of the triangle formed by the line $x + y = 4$, the x-axis, and the y-axis?

The figure shows the graphs of the equations $2*x + 3*y = 6$ and $4*x + 6*y = 12$. What can be concluded about the solutions to this pair of linear equations from the graph?

Two linear equations are represented by the lines $y = 2$ and $x = 3$. Based on the graph, which statement correctly describes the intersection of these two lines?

The graph of two linear equations $x - y = 0$ and $x + y = 2$ is shown below. What is the coordinates of the point of intersection of these two lines, and what does it represent?

A farmer has some hens and cows. If the total number of heads is $50$ and the total number of legs is $140$, let $x$ be the number of hens and $y$ be the number of cows. Plot $x + y = 50$ and $2x + 4y = 140$ to find how many cows the farmer has.

The area of a rectangle gets reduced by $9$ square units, if its length is reduced by $5$ units and breadth is increased by $3$ units. If we increase the length by $3$ units and the breadth by $2$ units, the area increases by $67$ square units. The resulting linear equations are $3x - 5y = 6$ and $2x + 3y = 61$. Solve graphically for length $x$ and breadth $y$.

A boat goes $30$ km upstream and $44$ km downstream in $10$ hours. In $13$ hours, it can go $40$ km upstream and $55$ km downstream. Let $u = 1/(v-s)$ and $v = 1/(v+s)$ where $v$ is boat speed and $s$ is stream speed. The linear equations are $30u + 44v = 10$ and $40u + 55v = 13$. Find the value of $u$ and $v$ graphically to determine the speed of the boat in still water.

How many solutions are there for the system $x + y = 4$ and $2x + 2y = 9$?

The pair of equations $2x + 3y = 5$ and $3x + 2y = 5$ has:

In the system $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$, if $\frac{a_1}{a_2} = \frac{b_1}{b_2}$, can the system have a unique solution?

The lines $\frac{x}{2} + \frac{y}{2} = 1$ and $x + y = 2$ are:

If the lines represented by the equations $(k-3)x + 3y = k$ and $kx + ky = 12$ are coincident, then the non-zero value of $k$ is:

For what value of $k$ will the system of equations $3x + (k-1)y = 1$ and $4x + (k+1)y = 5$ NOT have a unique solution?

If the system of linear equations $2x - (a-4)y = 2b+1$ and $4x - (a-1)y = 5b-1$ has infinitely many solutions, find the values of $a$ and $b$.

If the system $4x + 5y = 2$ and $(2a+7b)x + (a+8b)y = 2b-a+1$ has infinitely many solutions, find the relationship between $a$ and $b$.

If the lines $x + y = a$ and $x - y = b$ intersect at $(3, 1)$, what are the values of $a$ and $b$?

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

Solve the following pair of linear equations for $x$ and $y$: $x + 2y = 5$ and $x - y = -1$.

If $x + y = 18$ and $x - y = 6$, find the value of $y$.

Solve using elimination: $x + 2y = 10$ and $x - 2y = 2$.

Find $(x, y)$ if $x + 2y = 12$ and $x = y$.

Solve for $x$ and $y$: $2x - y = 5$ and $3x + 2y = 11$.

Find the values of $x$ and $y$: $5x + y = 13$ and $x - 3y = -7$.

Solve the pair of equations: $x + 4y = 14$ and $3x - y = 3$.

Solve for $x, y$: $\frac{xy}{x+y} = \frac{6}{5}$ and $\frac{xy}{y-x} = 6$.

Solve the system: $\frac{1}{3x+y} + \frac{1}{3x-y} = \frac{3}{4}$ and $\frac{1}{2(3x+y)} - \frac{1}{2(3x-y)} = -\frac{1}{8}$.

Determine $x$ and $y$: $\frac{x}{a} + \frac{y}{b} = a+b$ and $\frac{x}{a^2} + \frac{y}{b^2} = 2$.