# 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. Modify the equations so that the 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

### The Substitude method

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

Example 2 Solve variable y from the first equation Now we can substitute it into the other equation and solve x And so the variable for 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 to a pair of linear equations is the intersection of these lines (2,3), i.e. x = 2 and y = 3.

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