Solve Linear Combinations

Linear combination is combination of two lines. Linear combination is also called as addition method.  Linear combination is one of the methods to solve system of two equations. By this method, we can solve the variable x and y.

I like to share this Multiple Linear Regression Model with you all through my article.

Steps to solve 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.

Let us see about how to solve linear equations using linear combination method.

Linear Combination - Example Problem 1:

Solve the linear equations using linear combination method.

X – 2y = -2 and -3x – 6y = -6

Solution:

The given equations are

X – 2y = -2     (1)

-3x – 6y = -6  (2)

Multiply by 3 to the equation (1)

3x – 6y = -6

Multiply by 1 to the equation (2)

-3x – 6y = -6

Add both these equations.

3x – 6y + (-3x – 6y) = -6 + (-6)

3x – 6y – 3x – 6y = - 6 – 6

Combine like terms.

3x – 3x – 6y – 6y = - 12

-12y = - 12

Isolate the variable y.

Y = 1

Substitute the value of y variable into the equation (1)

X – 2y = - 2

X – 2(1) = - 2

X – 2 = - 2

Add 2 to each side.

X – 2 + 2 = - 2 + 2

X = 0

Therefore, the solutions are x = 0 and y = 1.

Linear Combination - Example Problem 2:

Solve the linear equations using linear combination method.

-3x + 3y = 6 and 6x + 9y = 12

Solution:

The given equations are

-3x + 3y = 6     (1)

6x + 9y = 12    (2)

Multiply by 2 to the equation (1)

-6x + 6y = 12

Multiply by 1 to the equation (2)

6x + 9y = 12

Add both these equations.

-6x + 6y + (6x + 9y) = 12 + 12

Combine like terms.

-6x + 6x + 6y + 9y = 24

15y = 24

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Isolate the variable y.

Y = `24/15`

Simplify it.

Y = `8/5`

Substitute the value of y variable into the equation (1)

-3x + 3y = 6

-3x + 3`(8/5)` = 6

-3x + `24/5` = 6

Multiply by 5 to each side.

-15x + 24 = 30

Subtract  24 to each side.

-15x + 24 - 24 = 30 – 24

-15x = 6

Isolate the variable x

X = `-6/15`

Simplify it.

X = `-2/5`

Therefore, the solutions are x = `-2/5` and y = `8/5` .