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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

- Identify the coefficient(s) and constant(s) in the following equations
- $3x – 4y + 7 = 0$
- $2x + 5y – 8 = 0$
- $x = 5$
- $2y = -7$

- Represent the following equations graphically.
- $2x – 4y + 8 = 0$
- $3x + 5y – 15 = 0$
- $2x + 6y – 12 = 0$
- $6x – 9y + 36 = 0$

- 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

### What is meant by linear equations in two variables?

A linear equation is an equation with degree 1. A linear equation in two variables is a type of linear equation in which there are two variables.

For example, $x – 2y = 4$, $2x + 3y =14$, etc. are linear equations in two variables.

### How do you identify linear equations in two variables?

We can identify a linear equation in two variables if it is expressed in the form $ax + by + c = 0$, consisting of two variables $x$ and $y$, and the highest degree of the given equation is $1$.

### How many solutions does a linear equation in two variables have?

A linear equation in two variables has an infinite number of solutions. All the points lying on the line $ax + by + c = 0$ are the solutions of the equation $ax + by + c = 0$.

### How will you check whether a given point is a solution of a linear equation in two variables?

If substituting the values of $x$ and $y$ from a point $(x, y)$ in the equation $ax + by + c = 0$ makes LHS and RHS, it means that $(x, y)$ is the solution of $ax + by + c = 0$, otherwise not.

For example, consider the equation $3x + 5y – 30 = 0$.

A point $\left(5, 3 \right)$ is the solution of the equation $3x + 5y + 30 = 0$, because $3 \times 5 + 5 \times 3 – 30 = 0$, whereas $\left(3, 5 \right)$ is not the solution of the equation $3x + 5y + 30 = 0$, because $3 \times 3 + 5 \times 5 – 30 = 9 + 25 – 30 = 4$, which is not equal to the RHS of the equation.

## 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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