How to Solve Linear Equations with 2 Variables – Tutorial and Practice

A system of a linear equations have two or more equations and two variables.   In a system of linear equations, each equation is a straight line and the solution will be the point where the two lines intersect.

If we have 2 or more linear equations with 2 or more variables, then we have a system of linear equations. Here, we will solve systems with 2 variables, given in 2 linear equations. Idea here is to express one variable using the other variable in one equation, and use it in the second equation, where we would get a linear equation with one variable. Let’s the how it works in one simple example:

x – y = 3

2x + y = 9

y = x – 3

2x + x – 3 = 9

3x = 9 + 3

From the first equation, we express y using x.

In the second equation, we write x – 3 instead of y. And there we get a linear equation with one variable x.

3x = 12

x = 12/3

x = 4

Now that we found x, we can use it to find y.

So, the solution of this system is (x,y) = (4,1)

y = x – 3

y = 4 – 3

y = 1

Let’s solve one more system using a different method:

5x – 3y = 17

x + 3y = 11

Notice that we have -3y in the first equation and +3y in the second. If we add these 2, we get zero, which means we lose variable y. So, we add these 2 equations and we get a linear equation with one variable.

5x – 3y + x + 3y = 17 – 11

6x = 6

x = 1

Now that we have x, we use it to find y.

5 – 3y = 17

-3y = 17 – 5

-3y = 12

y = 12/(-3)

y = -4

1. Solve the system:

1) (3,2)
2) (3,3)
3) (2,3)
4) (2,2)

2. Solve the system:

1)   (0, -?5)
2)   (0, ?5)
3) (-?5, 0)
4) (?5, 0)

3. Find x and y from the following system of equations:

1) (1,3)
2) (2,1)
3) (1,1)
4) (0,1)

4. Find x and y from the following system of equations:

2x+3=y+6
-4x-12=-8y

1) (3,2)
2) (1,3)
3) (3,3)
4) (2,2)

5. Solve the system, if a is some real number:

1. 3) (2,3)

2.  1)    (0, -?5)

3.  3) (1,1)

4. 2) (3,3)

5.  3)