# A Pair of Linear Equations in Two Variables

In a pair of linear equations in two variables, we have two equations and two variables. The solution to a pair of linear equations is also the solution of both equations.

The addition method always has the same steps

1. Move variables to the left and constants to the right

2. Select one of the variables

3. We modify the equations so that coefficients for the selected variable are opposite numbers.

5. Solve the remaining variable

6. Place the resolved variable into one of the original equations

Example 1

Solve a pair of linear equations using the addition method.

Let's choose y and multiply the lower equation by 2

We substitute x = 4 into the first equation

The solution to a pair of linear equations in two variables is x = 4 and y = 2

### Substitude method

We can also solve either variable from one equation and substitute it into the other equation.

Example 2

Solve variable y from the first equation

Now we can substitute it the other equation and solve for x

And so the variable y is

The solution is x = 2 and y = 3

### Solution by using graphs

We can also solve Example 2 by drawing graphs

The graphs are straight lines and the solution of a pair of linear equations is the intersection of these lines (2,3), i.e. x = 2 and y = 3.

### System of equations

In a system of equations there are multiple variables and equations. If there are three variables, three equations are needed to solve the variables.

Example 3

Solve the system of equations

First, add the first and second equations, so the y is cancelled out

Multiply the first equation by 2 and add it and the third equation to again cancel out the y

Now we have pair of linear equations

The solution to the pair of equations is x = 1 and z = 3. When x and z are substituted into the first equation of the system of equations, y = 2 is obtained.

The solution to the system of equations should be checked. By substituting x = 1, y = 2 and z = 3 in each equation of the system of equations, we find that the solution works for each equation.

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