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Linear Combination

To solve a given system of equation the linear combination method is used. Linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results or it is a result of combination of two lines. Linear combination is also called as addition method and is used to solve system of two equations. It is a modified form of elimination of variables. If the system has two unknowns, one of them is eliminated so as to get an equation of single unknown which can be solved easily.

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Linear Combination Method

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Linear combination is a method to solve a system algebraically. The goal of solving system is to reduce the system that has two equations in two variables to a single equation that has only one variable. In linear combination, the two equations are added together, resulting in an equation containing only one unknown and easily we can solve the system.

Solving Linear Systems by Linear Combinations

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Below you could see some steps to solve linear system:

Steps for solving systems of equation using linear combination method:

  • We need to add two line equations for eliminating any one variable.
  • Then we get a new equation of one variable.
  • Solve this, we get a value of that variable.
  • Substitute this variable value into any one original equation, we get a value for another variable.

Linear Combination Examples

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Below you could see examples for solving linear systems by linear combinations.

Solved Examples

Question 1: Solve the following linear equations using linear combination method.

7x - y = 5 and 2x + 3y = 8


Solution:

The given equations are

7X - y = 5 .........................(1)

2x + 3y = 8 ......................... (2)

Multiply by 3 to the equation (1)

21x - 3y = 15 .............................(3)

Multiply by 1 to the equation (2)

2x + 3y = 8 ............................. (4)

Add equation (3) and equation (4)

21x - 3y + (2x + 3y) = 15 + 8

21x - 3y + 2x + 3y = 15 + 8

Combine like terms.

21x + 2x - 3y + 3y = 23

23x = 23

Isolate the variable x.

x = 1

Substitute the value of x variable into the equation (2)

2x + 3y = 8

2(1) + 3y = 8

2 + 3y = 8

Subtract 2 from each side.

2 - 2 + 3y = 8 - 2

3y = 6

Isolate the variable y.

Y = 2

Therefore, the solutions are x = 1 and y = 2.



Question 2:

Solve the following linear equations using linear combination method.

3x - 2y = 10 and x + 2y = 6


Solution:

The given equations are

3x - 2y = 10 .......................(1)

x + 2y = 6 ....................... (2)

Here, coefficient of y in both the equations are opposite.

Adding both equations, we have

3x - 2y + (x + 2y) = 10 + 6

Combine like terms.

3x + x - 2y + 2y = 16

4x = 16

Isolate the variable x.

x = $\frac{16}{4}$

=> x = 4

Substitute the value of x variable into the equation (2)

x + 2y = 6

4 + 2y = 6

Subtract 4 from each side.

4 - 4 + 2y = 6 - 2

2y = 4

Isolate the variable y

y = $\frac{4}{2}$


=> y = 2

Therefore, the solutions are x = 4 and y = 2 .



Practice Problems

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Problem 1: Solve the given equations using linear combination method.
x - 3y = 1 and 5x + 2y = 5

Problem 2: Find the value of p and q using linear combination method.
a - 8b = 1 and 2a + 6b = 8

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