A linear equation with two variables are in the standard form of ax + by + c = 0, where x and y are variables and a, b, and c are real numbers and also a $\neq$ 0 and b $\neq$ 0. Example for two variable linear equation is 4x + 5y = 20.

A system of equations is a set of equations that involves two or more equations in two or more variables.

A solution of this system is an ordered pair that satisfies each equation in the system. Finding the set of all such solutions is called **solving the system of equations**. To solve the system of equations, we need to find the values of variables of equations.

**Methods for Solving**** the System of Equations:**

- Substitution Method
- Elimination Method
- Matrix Method
- Graphical Method

**Examples in Solving Systems of Equation:**

Let us see some example problems for solving systems of equation.

**Example 1:**

Solve the systems of equations by using the substitution method.

4x + 2y = 6

2x - 6y = 24

**Solution:**

**Step 1:**

**Given:** System of linear equations with two variables

4x + 2y = 6 --> (1)

2x - 6y = 24 --> (2)**Step 2: **

Multiply equation (2) by 2

4x - 12y = 48 -- (3)

Subtract equation (3) from (1)

4x + 2y = 6

-4x + 12y = -48

----------------------------

0 + 14y = -42

-----------------------------

=> 14y = -42

or y = -3

**Step 3:**

Substituting y = -3 in (1), we get

4x + 2y = 6

4x + 2 (- 3) = 6

4x - 6 = 6

4x = 12

or x = **3**

**Answer:** x = 3 and y = -3

**Example 2:**

Solve the system of equations by using the substitution method.

3a + b = 9

4a - b = 5

**Solution:**

**Step 1:**

System of linear equations with two variables

3a + b = 9 --> (1)

4a - b = 5 --> (2)

**Step 2:**

First take equations (1)

3a + b = 9

b = 9 – 3a

**Step 3:**

Substituting b = 9 – 3a in (2), we get

4a - b = 5

4a - (9 - 3a) = 5

4a - 9 + 3a = 5

4a + 3a = 5 + 9

7a = 14

a = 2

**Step 4:**

Substituting a = 2 in (1), we get

3a + b = 9

3(2) + b = 9

6 + b = 9

b = 9 - 6

b = 3

**Answer: **a = 2 and b = 3

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