# Linear Equations & Graphs

**Linear equation in one variable**

**Equation:**An equation is a statement of equality of two algebraic polynomial involving one or more variables.

**Linear equations:**The expression of the form Ax+B, where A and B are real numbers and A ≠ 0, is a linear polynomial and equation involving only linear polynomial are called as linear equations.

- Graph of linear equation of one variable is a straight line, which is either parallel to the horizontal and vertical axis.
- Linear equation in one variables has unique solution.

**linear equations examples:**

- y = 3x + 1
- 4x = 6 + 3y
- y/2 = 3 − x

**linear equations in one variable example:**

- 9x + 1 = 5 − x
- t + 3t = 9 − t

**Rules for solving a linear equation**

- If some number is added to both the sides of an equation, the equality remains the same.
- If same number is subtracted to both the sides of an equation, the equality remains the same.
- If same number is multiplied to both the sides of the equation, the equality remains the same.
- If both sides are divided by a some non-zero number the equality remains the same.

**Example:**Solve 2(x – 3)-(5 – 3x)=3(x + 1)-4(2 + x) and the value of x is ?

**Solution:**

2(x – 3)-(5 – 3x)=3(x + 1)-4(2 + x)

= 2x – 6 – 5 + 3x = 3x + 3 – 8 – 4x

= 5x – 11 = -x -5

= 6x = 6

= x = 1 ( Ans)

**Linear equation in two variable :**

An equation of the form

**ax + by + c = 0**, where a,b,c are real, a ≠ 0, b ≠ 0 and x and y are variables. It is called as linear equation in two variables.__2x + 3y = 5, 3√x − y = 6 are linear equation in two variables.__

**Example:****Important points:**

- The linear equation
**ax + by + c = 0**has an infinite number of solutions. - The graph of equation
**ax + by + c = 0**is a straight line so it is called as linear equations. - Every point on graph of
**ax + by + c = 0**gives its solution.

**Systems of linear equation in two variables.**

When we draw the graph of each of the two equations, we have the following three possibilities.

- Either the lines intersect ( Consistent )
- The lines are parallel ( Inconsistent )
- The lines coincide ( Dependent )

**Case 1 :**

**When two lines intersect**– As two intersecting straight lines always intersect in a point. So there is only one point which lies on both the lines.

If the straight lines in the graph intersect each other at a single point, then system has unique solution.

**Case 2 :**

**When two lines are parallel**– When two lines are parallel they do not meet at any point so there is no common point on both of the straight lines, So thus system has no solution.

**Case 3**:

**When the lines coincide-**When two lines coincide, the points lying on one also belong to the other. Thus all points are common so the system has infinite number of solution.

**Consistency of the system of linear equations:**

A set of linear equations is said to be consistent, if there exists at least one solution for these equation. A set of linear equations is said to be inconsistent, if there is no solution for these equation.

Let us consider a system of two linear equation as shown.

**The system of equations**

**a1x + b1y + c1 = 0**and

**a2x + b2y + c2 =**0

has

**Case 1**no solution if

**( Inconsistent )**

**a1/a2 = b1/b2 ≠ c1/c2**

**Case 2**an infinite number of solution, if

**(Dependent )**

**a1/a2 = b1/b2 = c1/c2**

**Case 3**a unique solution, if

**( Consistent )**

**a1/a2 ≠ b1/b2**

- A solution can be obtained by any of the above four stated method.

**linear equation problems:**

**Question :**Which option is correct, for the following pair of equations ?

**x + 2y – 4 = 0**and

**3x + 6y -12 = 0**

(a) Consistent

(b) Consistent (Dependent)

(c) Inconsistent

(d) None of these

**Solution:**

Given pair of linear equations is

**x + 2y – 4 = 0**and**3x + 6y -12 = 0**On comparing with standard form of pair of linear equations, we get

a1 = 1, b1 = 2, c1 = -4

and a2 = 3, b2 = 6, c2 = -12

Now, a1/a2 = 1/3, b1/b2 = 2/6 = 1/3, c1/c2 = -4/-12 = 1/3

Clearly,

**a1/a2 = b1/b2 = c1/c2**

Hence , the given pair of linear equations is

**consistent ( dependent )**, So option b is correct.

__Note- If you have any question then you can ask in comment box.__

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