• Home
• /
• Blog
• /
• Linear Equations in Two Variables – Definition, Types, and Graphs

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

November 1, 2022

This post is also available in: हिन्दी (Hindi)

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 $-9$.

Is your child struggling with Maths?
We can help!
Country
• Afghanistan 93
• Albania 355
• Algeria 213
• American Samoa 1-684
• Andorra 376
• Angola 244
• Anguilla 1-264
• Antarctica 672
• Antigua & Barbuda 1-268
• Argentina 54
• Armenia 374
• Aruba 297
• Australia 61
• Austria 43
• Azerbaijan 994
• Bahamas 1-242
• Bahrain 973
• Belarus 375
• Belgium 32
• Belize 501
• Benin 229
• Bermuda 1-441
• Bhutan 975
• Bolivia 591
• Bosnia 387
• Botswana 267
• Bouvet Island 47
• Brazil 55
• British Indian Ocean Territory 246
• British Virgin Islands 1-284
• Brunei 673
• Bulgaria 359
• Burkina Faso 226
• Burundi 257
• Cambodia 855
• Cameroon 237
• Cape Verde 238
• Caribbean Netherlands 599
• Cayman Islands 1-345
• Central African Republic 236
• Chile 56
• China 86
• Christmas Island 61
• Cocos (Keeling) Islands 61
• Colombia 57
• Comoros 269
• Congo - Brazzaville 242
• Congo - Kinshasa 243
• Cook Islands 682
• Costa Rica 506
• Croatia 385
• Cuba 53
• Cyprus 357
• Czech Republic 420
• Denmark 45
• Djibouti 253
• Dominica 1-767
• Egypt 20
• Equatorial Guinea 240
• Eritrea 291
• Estonia 372
• Ethiopia 251
• Falkland Islands 500
• Faroe Islands 298
• Fiji 679
• Finland 358
• France 33
• French Guiana 594
• French Polynesia 689
• French Southern Territories 262
• Gabon 241
• Gambia 220
• Georgia 995
• Germany 49
• Ghana 233
• Gibraltar 350
• Greece 30
• Greenland 299
• Guam 1-671
• Guatemala 502
• Guernsey 44
• Guinea 224
• Guinea-Bissau 245
• Guyana 592
• Haiti 509
• Heard & McDonald Islands 672
• Honduras 504
• Hong Kong 852
• Hungary 36
• Iceland 354
• India 91
• Indonesia 62
• Iran 98
• Iraq 964
• Ireland 353
• Isle of Man 44
• Israel 972
• Italy 39
• Jamaica 1-876
• Japan 81
• Jersey 44
• Jordan 962
• Kazakhstan 7
• Kenya 254
• Kiribati 686
• Kuwait 965
• Kyrgyzstan 996
• Laos 856
• Latvia 371
• Lebanon 961
• Lesotho 266
• Liberia 231
• Libya 218
• Liechtenstein 423
• Lithuania 370
• Luxembourg 352
• Macau 853
• Macedonia 389
• Malawi 265
• Malaysia 60
• Maldives 960
• Mali 223
• Malta 356
• Marshall Islands 692
• Martinique 596
• Mauritania 222
• Mauritius 230
• Mayotte 262
• Mexico 52
• Micronesia 691
• Moldova 373
• Monaco 377
• Mongolia 976
• Montenegro 382
• Montserrat 1-664
• Morocco 212
• Mozambique 258
• Myanmar 95
• Namibia 264
• Nauru 674
• Nepal 977
• Netherlands 31
• New Caledonia 687
• New Zealand 64
• Nicaragua 505
• Niger 227
• Nigeria 234
• Niue 683
• Norfolk Island 672
• North Korea 850
• Northern Mariana Islands 1-670
• Norway 47
• Oman 968
• Pakistan 92
• Palau 680
• Palestine 970
• Panama 507
• Papua New Guinea 675
• Paraguay 595
• Peru 51
• Philippines 63
• Pitcairn Islands 870
• Poland 48
• Portugal 351
• Puerto Rico 1
• Qatar 974
• Romania 40
• Russia 7
• Rwanda 250
• Samoa 685
• San Marino 378
• Saudi Arabia 966
• Senegal 221
• Serbia 381 p
• Seychelles 248
• Sierra Leone 232
• Singapore 65
• Slovakia 421
• Slovenia 386
• Solomon Islands 677
• Somalia 252
• South Africa 27
• South Georgia & South Sandwich Islands 500
• South Korea 82
• South Sudan 211
• Spain 34
• Sri Lanka 94
• Sudan 249
• Suriname 597
• Svalbard & Jan Mayen 47
• Swaziland 268
• Sweden 46
• Switzerland 41
• Syria 963
• Sao Tome and Principe 239
• Taiwan 886
• Tajikistan 992
• Tanzania 255
• Thailand 66
• Timor-Leste 670
• Togo 228
• Tokelau 690
• Tonga 676
• Tunisia 216
• Turkey 90
• Turkmenistan 993
• Turks & Caicos Islands 1-649
• Tuvalu 688
• U.S. Outlying Islands
• U.S. Virgin Islands 1-340
• UK 44
• US 1
• Uganda 256
• Ukraine 380
• United Arab Emirates 971
• Uruguay 598
• Uzbekistan 998
• Vanuatu 678
• Vatican City 39-06
• Venezuela 58
• Vietnam 84
• Wallis & Futuna 681
• Western Sahara 212
• Yemen 967
• Zambia 260
• Zimbabwe 263
• Less Than 6 Years
• 6 To 10 Years
• 11 To 16 Years
• Greater Than 16 Years

## 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

### 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.