**Equation**

An equation is a statement that two mathematical expressions having one or more variables are equal.

**Linear Equation**

Equations in which the powers of all the variables involved are one are called linear equations. The degree of a linear equation is always one.

**General form of a Linear Equation in Two Variables**

The general form of a linear equation in two variables is ax + by + c = 0, where a and b cannot be zero simultaneously.

**Representing linear equations for a word problem**

To represent a word problem as a linear equation

- Identify unknown quantities and denote them by variables.
- Represent the relationships between quantities in a mathematical form, replacing the unknowns with variables.

**Solution of a Linear Equation in 2 variables**

The solution of a linear equation in two variables is a pair of values, one for *x* and the other for y, which makes the two sides of the equation equal.

Eg: If 2*x*+y=4, then (0,4) is one of its solutions as it satisfies the equation. A linear equation in two variables has infinitely many solutions.

**Geometrical Representation of a Linear Equation**

Geometrically, a linear equation in two variables can be represented as a straight line.

2*x* – y + 1 = 0

⇒ y = 2*x* + 1

Graph of y = 2*x*+1

Plotting a Straight Line

The graph of a linear equation in two variables is a straight line. We plot the straight line as follows:

Any additional points plotted in this manner will lie on the same line.

**All about Lines**

General form of a pair of linear equations in 2 variables A pair of linear equations in two variables can be represented as follows

The coefficients of x and y cannot be zero simultaneously for an equation.

**Nature of 2 straight lines in a plane**

For a pair of straight lines on a plane, there are three possibilities

i) They intersect at exactly one point

pair of linear equations which intersect at a single point.

ii) They are parallel

pair of linear equations which are parallel.

iii) They are coincident

pair of linear equations which are coincident.

**Graphical Solution**

Representing pair of LE in 2 variables graphically

Graphically, a pair of linear equations in two variables can be represented by a pair of straight lines.

Graphical method of finding solution of a pair of Linear Equations

Graphical Method of finding the solution to a pair of linear equations is as follows:

- Plot both the equations (two straight lines)
- Find the point of intersection of the lines.

The point of intersection is the solution.

Comparing the ratios of coefficients of a Linear Equation

**Algebraic Solution**

Finding solution for consistent pair of Linear Equations

The solution of a pair of linear equations is of the form (x,y) which satisfies both the equations simultaneously. Solution for a consistent pair of linear equations can be found out using

i) Elimination method

ii) Substitution Method

iii) Cross-multiplication method

iv) Graphical method

**Substitution Method of finding solution of a pair of Linear Equations**

Substitution method:

y – 2*x* = 1

*x* + 2y = 12

(i) express one variable in terms of the other using one of the equations. In this case, y = 2*x* + 1.

(ii) substitute for this variable (y) in the second equation to get a linear equation in one variable, x. *x* + 2 × (2*x* + 1) = 12

⇒ 5 *x* + 2 = 12

(iii) Solve the linear equation in one variable to find the value of that variable.

5 *x* + 2 = 12

⇒ *x* = 2

(iv) Substitute this value in one of the equations to get the value of the other variable.

y = 2 × 2 + 1

⇒y = 5

So, (2,5) is the required solution of the pair of linear equations y – 2*x* = 1 and *x* + 2y = 12.

**Elimination method of finding solution of a pair of Linear Equations**

**Elimination method**Consider

*x*+ 2y = 8 and 2

*x*– 3y = 2

**Step 1:** Make the coefficients of any variable the same by multiplying the equations with constants. Multiplying the first equation by 2, we get,

2*x* + 4y = 16**Step 2:** Add or subtract the equations to eliminate one variable, giving a single variable equation.

Subtract second equation from the previous equation

2x + 4y = 16

2x – 3y = 2

– + –

———————–

0(x) + 7y =14**Step 3:** Solve for one variable and substitute this in any equation to get the other variable.

y = 2,

x = 8 – 2 y

⇒ x = 8 – 4

⇒ x = 4

(4, 2) is the solution.

**Cross-multiplication Method of finding solution of a pair of Linear Equations**

For the pair of linear equations

a_{1}*x *+ b_{1}*y *+ c_{1}=0

a_{2}x + b_{2}y + c_{2}=0,

x and y can be calculated as

x = (b_{1}c_{2}−b_{2}c_{1})/(a_{1}b_{2}−a_{2}b_{1})

y = (c_{1}a_{2}−c_{2}a_{1})/(a_{1}b_{2}−a_{2}b_{1})

Solving Linear Equations

**Equations reducible to a pair of Linear Equations in 2 variables**

Some equations may be in a form which can be reduced to a linear equation through substitution.

2/x+3/y=4

5/x−4/y=9

In this case, we may make the substitution

1/x = u and 1/y = v

The pair of equations reduces to

2u + 3v = 4

5u – 4v = 9

The above pair of equations may be solved. After solving, back substitute to get the values of x and y.