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# Solving Systems of Equations

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 equationsTo 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

 Related Calculators Solve a System of Equations Calculator

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