Linear Equations in Two Variables – Definition, Types, and Graphs

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Linear equations in one variable of the form $ax + b = 0$ are the equations that have a unique solution (or only one solution) and the solution of such equations consists of a single number that can be represented on a number line. On the other hand, linear equations in two variables are the equations of the form $ax + by + c = 0$. The solution of such equations is an ordered pair of the form $\left(x, y \right)$, where $x$ denotes $x$-coordinnate and $y$ denotes the $y$-coordinate on a Cartesian plane.

Let’s understand what are linear equations in two variables and what are its different types and its solutions with examples.

What are Linear Equations in Two Variables?

An equation is called a linear equation in two variables if it is written in the form of $ax + by + c =0$, where $a$, $b$, and $c$ are real numbers and the coefficients of the variables $x$ and $y$, i.e $a$ and $b$ respectively, are not equal to zero.

For example, $2x + 5y = 7$ and $-x + 3y = 5$ are linear equations in two variables.

The solution for such an equation is a pair of values (ordered pair), one for $x$ and one for $y$ which makes the two sides (LHS and RHS)of an equation equal. The solutions of a linear equation in two variables can be represented on a Cartesian plane, i.e., $xy$-plane.

Examples

Let’s understand the variables and coefficients in linear equations in two variables with a few examples.

Ex 1: What are the variables in the equation $-3m + 2n + 7 = 0$?

The equation $-3m + 2n + 7 = 0$ is of the form $ax + by + c = 0$.

Comparing the equation $-3m + 2n + 7 = 0$ with $ax + by + c = 0$, we get

$x = m$ and $y = n$, therefore, the variables are $m$ and $n$, or we can say that $-3m + 2n + 7 = 0$ is a linear equation in two variables with variables $m$ and $n$.

Ex 2: Identify the coefficient(s) and constant(s) in the equation $7x + 2y + 8 = 0$.

The equation $7x + 2y + 8 = 0$ is of the form $ax + by + c = 0$.

Comparing the equation $7x + 2y + 8 = 0$ with $ax + by + c = 0$, we get $a = 7$, $b = 2$ and $c = 8$.

Therefore, the coefficients of the equation $7x + 2y + 8 = 0$ are $7$ and $2$, $7$ is the coefficient of $x$ and $2$ is the coefficient of $y$.

And, the constant in the equation $7x + 2y + 8 = 0$ is $8$.

Ex 3: Identify the coefficient(s) and constant(s) in the equation $5x – 3y = 9$.

The equation $5x – 3y = 9$ can be written as $5x – 3y – 9 = 0$ (Shifting $9$ from RHS to LHS).

Now, the equation $5x – 3y – 9 = 0$ is of the form $ax + by + c = 0$.

Comparing the equation $5x – 3y – 9 = 0$ with $ax + by + c = 0$, we get $a = 5$, $b = -3$ and $c = -9$.

Therefore, the coefficients of the equation $5x – 3y = 9$ are $5$ and $-3$, $5$ is the coefficient of $x$ and $-3$ is the coefficient of $y$.

And, the constant in the equation $5x – 3y = 9$ is $-9$.

Graph of a Linear Equation in Two Variables

A linear equation in two variables can be represented as a straight line on a Cartesian plane, i.e., $xy$-coordinate axes. Since the graph is a straight line (linear), therefore, the equations of the form $ax + by + c = 0$ are called linear equations and as there are two variables in it, hence the name ‘linear equation in two variables’.

To plot the graph of a linear equation $ax + by + c = 0$, we consider some arbitrary values of $x$ and get their corresponding $y$ values and then plot those points on a Cartesian plane. These points are then joined by a straight line which represents the equation $ax + by + c = 0$.

Note: To plot a straight line, only two points are enough.

Examples

Let’s consider an example to understand how a graph of a linear equation in two variables is plotted.

Ex 1: Represent the equation $3x + 4y – 12 = 0$ graphically.

$3x + 4y – 12 = 0 => 3x + 4y = 12$

Substituting $x = 0$ in the equation, we get  $0 + 4y = 12 => y = \frac{12}{4} => y = 3$.

Therefore, one point lying on the graph of $3x + 4y – 12 = 0$ is $(0, 3)$.

Substituting $y = 0$ in the equation, we get  $3x + 0 = 12 => x = \frac{12}{3} => x = 4$.

Therefore, another point lying on the graph of $3x + 4y – 12 = 0$ is $(4, 0)$.

Now, plotting the points $\left(0, 3 \right)$ and $\left(4, 0 \right)$ on a Cartesian plane, we get a straight line representing the equation $3x + 4y – 12 = 0$.

Graph of Horizontal Line

The general equation of a linear equation in two variables is of the form $ax + by + c = 0$, where $a$ and $b$ are coefficients of the variables $x$ and $y$ respectively and $c$ is a constant. Further $a$, $b$, and $c$ are real numbers. When the coefficient $a = 0$, then the equation reduces to the form $by + c = 0$, which can also be written as $by = -c$. This equation represents a straight horizontal line parallel to $x$-axis and perpendicular to $y$-axis and crossing the $y$-axis at a point $\left(0, -\frac {c}{b} \right)$. The point $\left(0, -\frac {c}{b} \right)$ is called the $y$-intercept of a line $by + c = 0$.

Example

Ex 1: Represent the equation $2y – 18 = 0$.

$2y – 18 = 0 => 2y = 18 => y = \frac{18}{2} => y = 9$

Therefore, the line $2y – 18 = 0$ passes through a point $\left(0, 9 \right)$ which is parallel to $x$-axis and perpendicular to $y$-axis. The line has $y$-intercept $\left(0, 9\right)$.

Graph of Vertical Line

The general equation of a linear equation in two variables is of the form $ax + by + c = 0$, where $a$ and $b$ are coefficients of the variables $x$ and $y$ respectively and $c$ is a constant. Further $a$, $b$, and $c$ are real numbers. When the coefficient $b = 0$, then the equation reduces to the form $ax + c = 0$, which can also be written as $ax = -c$. This equation represents a straight horizontal line parallel to $y$-axis and perpendicular to $x$-axis and crossing the $x$-axis at a point $\left(-\frac {c}{a}, 0 \right)$. The point $\left(-\frac {c}{a}, 0 \right)$ is called the $x$-intercept of a line $ax + c = 0$.

Example

Ex 1: Represent the equation $3x – 15 = 0$.

$3x – 15 = 0 => 3x = 15 => x = \frac{15}{3} => x = 5$

Therefore, the line $3x – 15 = 0$ passes through a point $\left(5, 0 \right)$ which is parallel to $y$-axis and perpendicular to $x$-axis. The line has $x$-intercept $\left(5, 0\right)$.

Graph of Slanted Line

The general equation of a linear equation in two variables is of the form $ax + by + c = 0$, where $a$ and $b$ are coefficients of the variables $x$ and $y$ respectively and $c$ is a constant. Further $a$, $b$, and $c$ are real numbers. When both the coefficients $a$ and $b$ are non-zeroes, then the graph of the equation $ax + by + c = 0$ will be a slanted line.

Examples

Ex 1: Represent the equation $3x + 4y – 12 = 0$.

$3x + 4y – 12 = 0 => 3x + 4y = 12$

Substituting $x = 0$ in the equation, we get  $0 + 4y = 12 => y = \frac{12}{4} => y = 3$.

Therefore, one point lying on the graph of $3x + 4y – 12 = 0$ is $(0, 3)$.

Substituting $y = 0$ in the equation, we get  $3x + 0 = 12 => x = \frac{12}{3} => x = 4$.

Therefore, another point lying on the graph of $3x + 4y – 12 = 0$ is $(4, 0)$.

Now, plotting the points $\left(0, 3 \right)$ and $\left(4, 0 \right)$ on a Cartesian plane, we get a straight line representing the equation $3x + 4y – 12 = 0$.

Solution of Linear Equations in Two Variables

We learned that when a linear equation in two variables of the form $ax + by + c = 0$ is plotted in a Cartesian plane, we get a straight line. Any point lying on the line $ax + by + c = 0$ will be the solution of the equation $ax + by + c = 0$.

Since there are infinite points lying on the line $ax + by + c = 0$, therefore, there exist infinite (countless) solutions for any linear equation in two variables.

As there are two variables in a linear equation of the form $ax + by + c = 0$, a solution means a pair of values, one for $x$ and one for $y$ which satisfy the given equation. 

Let us consider the equation $2x + 3y = 12$. Here, $x = 3$ and $y = 12$ is a solution because when you substitute $x = 3$ and $y = 2$ in the equation, we get $2 \times 3 + 3 \times 2 = 12$, which is the RHS of the equation.

Similarly, when you substitute $x = 0$ and $y = 4$ in the equation, we get $2 \times 0 + 3 \times 4 = 12$, which is also the RHS of the equation. 

And also, when you substitute $x = 6$ and $y = 0$ in the equation, we get $2 \times 6 + 3 \times 0 = 12$, which again is the RHS of the equation. 

So, we see that all these points $\left(3, 2 \right)$, $\left(0, 4 \right)$, and $\left(6, 0 \right)$ are the solutions of the equation $2x + 3y = 12$.

In fact, there are infinite solutions for the equation $2x + 3y = 12$ and in general for any linear equation of the form $ax + by = c$.

Equations of $x$ and $y$ Axes

We learned above that any

• linear equation of the form $ax + c = 0$ is parallel to $y$-axis or perpendicular to $x$-axis
• linear equation of the form $by + c = 0$ is parallel to $x$-axis or perpendicular to $y$-axis

When $c$ becomes $0$ in the equation $ax + c = 0$, we get $ax = 0 => x = \frac {0}{a} => x = 0$, which is the equation of $y$-axis.

Similarly, when $c$ becomes $0$ in the equation $by + c = 0$, we get $by = 0 => y = \frac {0}{b} => y = 0$, which is the equation of $x$-axis.

Difference Between Linear Equations in One Variable and Two Variables

These are the differences between linear equations in one variable and linear equations in two variables.

Practice Problems

1. Identify the coefficient(s) and constant(s) in the following equations
   • $3x – 4y + 7 = 0$
   • $2x + 5y – 8 = 0$
   • $x = 5$
   • $2y = -7$
2. Represent the following equations graphically.
   • $2x – 4y + 8 = 0$
   • $3x + 5y – 15 = 0$ 
   • $2x + 6y – 12 = 0$
   • $6x – 9y + 36 = 0$
3. Find any four solutions for the following equations.
   • $x – y – 8 = 0$
   • $4x + 3y + 14 = 0$
   • $7x – 3y – 21 = 0$
   • $5x + 2y + 20 = 0$

FAQs

Conclusion

The equations of the form $ax + by + c = 0$ are called linear equations in two variables and such equations have infinite solutions. All the points lying on the line represented by $ax + by + c = 0$ are the solutions of the equation.

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