Linear combination is a set of terms where each term is multiplied by a constant and the results are added to it. Linear combination place an very important role in the field of mathematics and is central to linear algebra.

Let $v_{1},v_{2},....,v_{r}$ be vectors in $R^{n}$. A linear combination of these vectors is of the form $k_{1}v_{1}$ + $k_{2}v_{2}$ + ......+ $k_{r}v_{r}$
The coefficients $k_{1},k_{2},.....,k_{r}$ are scalars.

Definition of Linear Combination

Linear combination is a sum of products of each quantity times a constant.

For example, the expression aX + bY + cZ is a linear combination of X, Y, and Z where a, b, c are constants. A system of linear equations is one which may be written in the form:

$a_{11}x_{1}$ + $a_{12}x_{2}$ + .........+ $a_{1n}x_{n}$ = $b_{1}$
$a_{21}x_{1}$ + $a_{22}x_{2}$ + .........+ $a_{2n}x_{n}$ = $b_{2}$
...... ........
....... .......
$a_{m1}x_{1}$ + $a_{m2}x_{2}$ + .........+ $a_{mn}x_{n}$ = $b_{m}$

All of the coefficients $a_{ij}$ and $b_{i}$ are assumed to be known constants. All the $x_{i}$'s are assumed to be unknowns that we are to solve for.
Linear combination will have two or more linear equations. If two linear equations intersect then the point of intersection is called the solution to the system of linear equations.

Solving Linear Systems by Linear Combinations

Steps for solving linear systems by linear combinations:

1. Move all terms with variables to one side.
2. From the given set of equations choose the variable to move first.
3. Since number of equations should be equal to number of variables multiply the equations if necessary (Opposite equations).
4. Add the equations and solve for the variables which are remaining.
5. Solve for the eliminated variable, once solved you will have the solution.

Linear Combination Examples

Solved Examples

Question 1: Solve the system:
x + y = 22
x - y = 15

Solution:

The given equations are:
x + y = 22
x - y = 15
From the given set of equations we see that the number of equations is equal to the number of variables.
There is no need to multiply as they can be  solved simultaneously.

x + y = 22

x - y = 15
__________
2x = 37
___________

$\Rightarrow$  x = $\frac{37}{2}$
x = 18.5
Substitute x = 18.5 in any of the given equations
We substitute in x + y = 22
$\Rightarrow$    18.5 + y = 22
$\Rightarrow$       y = 22 - 18.5
$\Rightarrow$   y = 3.5

Therefore the values of x and y are 18.5 and 3.5 respectively.

Question 2: Solve the system using linear combination.
3x - 2y = 7
2x + 4y = 10

Solution:

In the given problem we see that  all the variables are on one side.
We choose to eliminate y first from the given set of equations. So we multiply the first equation by 2.
(3x - 2y = 7) $\times$ 2
$\Rightarrow$  6x - 4y = 14

6x - 4y = 14
2x + 4y = 10
____________
8x = 24
____________
$\Rightarrow$   x = 3

Now substitute x = 3 in 3x - 2y = 7 we get,
3(3) - 2y =7
9 - 2y = 7
-2y = 7 - 9
$\Rightarrow$    -2y = -2
So  we get y = 1

Therefore x = 3  and y = 1 are the values for the given set of equations.