Linear Equations (Types and Solved Examples)

What is Linear Equation?

A linear equation is an algebraic equation in which the highest exponent of the variable is one. Linear equation has one, two or three variables but not every linear system with 03 equations. Usually, a system of linear equation has only a single solution but sometimes, it has no solution or infinite number of solutions.

A two variables linear equation describes a relationship in which the value of one variable say ‘x’ depend on the value of the other variable say ‘y’. If there are two variables, the graph of a linear equation will be straight line.

Standard Form of Linear Equation

Linear Equations have a standard form like:

Ax + By = C

Here, A, B, and C are coefficients, whereas, x and y are variables.

General form of the linear equation with two variables is:

y = mx + c, m ≠ 0

Linear Equation Formula

Some general formulas are:

Slop Intercept Form : $\displaystyle y=mx+c$
Point Form: $\displaystyle y-{{y}_{1}}=m({{x}_{1}}-{{x}_{2}})$
Two Point Form: $\displaystyle \frac{{y-{{y}_{1}}}}{{{{y}_{2}}-{{y}_{1}}}}=\frac{{x-{{x}_{1}}}}{{{{x}_{2}}-{{x}_{1}}}}$

Examples of Linear Equations

$\displaystyle \begin{array}{l}1.\,\,\,10x-20=0\\2.\,\,\,y-12=3\\3.\,\,\,\frac{x}{2}-\frac{y}{2}=12\\4.\,\,\,x+2y=20\\5.\,\,\,x+y+z-45=9\end{array}$

In above examples, the highest exponent of the variable is 1.

Equation with one Variable: An equation having one variable, e.g.
12x – 10 = 0
12x = 10
Equation with two Variables: An equation having two variables, e.g.
12x +10y – 10 = 0
12x +23y = 20
Equation with three Variables: An equation having three variables, e.g.
12x +10y -3z – 10 = 0
12x +23y – 12z = 20

Solved Examples of Linear Equations:

Example No.1:

$\displaystyle 12x+10=4x+26$

Solution:

$\displaystyle \begin{array}{l}12x+10=4x+26\\12x-4x=26-10\\8x=16\\x=\frac{{16}}{8}\\x=2\end{array}$

Example No.2:

$\displaystyle x=25-4x$

Solution:

$\displaystyle \begin{array}{l}x=25-4x\\5x=25\\x=\frac{{25}}{5}\\x=5\end{array}$

Example No.3:

$\displaystyle \frac{4}{{x+9}}=\frac{2}{{x+4}}$

Solution:

$\displaystyle \begin{array}{l}\frac{4}{{x+9}}=\frac{2}{{x+4}}\\4(x+4)=2(x+9)\\4x+16=2x+18\\4x-2x=18-16\\2x=2\\x=1\end{array}$

In linear equation, the sign of equality (=) divides the equation into two sides such as L.H.S. and R.H.S.

In the given equation, the value of the variable which makes L.H.S = R.H.S is called the solution of linear equation.

Examples No.1

x + 6 = 8 is a linear equation.

Here, L.H.S. is x + 6 and R.H.S. is 8

If we put x = 2, then left hand side will be 2 + 6 which is equal to right hand side

Thus, the solution of the given linear equation will be x = 2

Example No.2

3x – 2 = 2x – 3 is a linear equation

If we put x = -1, then left hand side will be 3(-1) – 2 and right hand side will be 2(-1) – 3

We obtained,

-3 – 2= -2 – 3

-5 = -5

Therefore, L.H.S. = R.H.S.

So, x = -1 is the solution of given linear equation.

Types of Linear Equation:

There are three types of linear equations

Conditional Equation
Identity Equation
Contradiction Equation

1. Conditional Equation:

Conditional equation has only one solution. For example,

$\displaystyle \begin{array}{l}4(x-1)+8=2x+10\\4x-4+8=2x+10\\4x+4=2x+10\\4x-2x=10-4\\2x=6\\x=\frac{6}{2}\\x=3\end{array}$

2. Identity Equation:

An identity equation is always true and every real number is a solution of it, therefore, it has infinite solutions. The solution of a linear equation which has identity is usually expressed as

$\displaystyle x\in R$

Sometimes, left hand side is equal to the right hand side (probably we obtain 0=0), therefore, we can easily find out that this equation is an identity. For example,

$\displaystyle \begin{array}{l}4(x-5)+4=x+3(x+2)-22\\4x-20+4=x+3x+6-22\\4x-16=4x-16\\4x-4x=16-16\\0=0\end{array}$

3. Contradiction Equation:

A Contradiction equation is always false and has no solution. Contradiction equation is mostly expressed as:

$\displaystyle x\in \varnothing$

For example,

$\displaystyle \begin{array}{l}4(x-2)+12=x+3(x+4)\\4x-20+12=x+3x+12\\4x-8=4x+12\\4x-4x=12+8\\0=16\end{array}$

Linear Equations represent lines

An equation represents a line on a graph and we have required two points to draw a line through those points. On a graph, ‘x’ and ‘y’ variables show the ‘x’ and ‘y’ coordinates of a graph. If we put a value for ‘x’ then we can easily calculate the corresponding value of ‘y’ and those two values will show a point on a graph. Similarly, if we keep putting the value of ‘x’ and ‘y’ in the given linear equation, we can obtain a straight line on the graph.

Graphical representation of Linear Equation

We can put the values of ‘x’ and ‘y’ into the equation in order to graph a linear equation. We can use the “intercept” points. Few below mentioned points must be follow:

Put x = 0 into the equation and solve for y and plot the point (0,y) on the y-axis
Put y = 0 into the equation and solve for x and plot the point (x,0) on the x-axis
Finally, draw a straight line between the two points

Check your skills to find the solutions of these linear equations:

$\displaystyle \begin{array}{l}\text{1}\text{.}\,\text{ }2x-11=3x+9\\2.\,\text{ }12y+\text{1}4=5-2y\\\text{3}\text{.}\,\text{ }12-3\left( {x-4} \right)=x+10\\~4.\,\text{ 1}5\left( {y-4} \right)=3\left( {2y-15} \right)-\left( {5-2y} \right)\\~5.\text{ }\,\text{3}\left( {x+21} \right)+6x=10-2\left( {x+4} \right)~\\\text{6}\text{. }\,\frac{x}{2}-\frac{{\left( {x-2} \right)}}{3}\text{= }\frac{7}{{30}}\\7.\text{ }\,\frac{{\left( {x-3} \right)}}{3}+\frac{{\left( {x-1} \right)}}{3}\text{+}\frac{{\left( {x-2} \right)}}{2}=1\\8.\text{ }\,\frac{{\left( {3y-2} \right)}}{3}+\frac{{\left( {2y+3} \right)}}{2}=\left( {y+7} \right)\\\text{9}\text{. }\,\frac{{\left( {8x-5} \right)}}{{4x-1}}=\frac{{-4}}{{10}}\\\text{10}\text{. }\,\frac{{\left( {4-7y} \right)}}{{2+3y}}=\frac{{-8}}{7}\end{array}$