Systems of Equations (Two Variables, Two Equations)

A system of equations is nothing but a collection of more than one equation that comes with the same set of unknowns. In order to solve it, we need to find values for every unknown variable that will help in satisfying the system of equations.

If you have an equation like ax+by+c=0, then here it can have a, b and c as real numbers and a and b cannot be equal to zero. In order to find a solution for that equation, you need to find values for both x and y so that it will make the two sides equal.

Let us take an example of linear equations in two variables and understand:

Solve the following system of linear equations:

Equation 1: -4 = 2x – 1y

Equation 2: -2 = 3x – 1y

There are three methods to solve the system of equations: graphing, substitution method, or the elimination method. In this article, we will focus on the substitution and graphing method.

Substitution Method

1. Take the first equation and isolate one of the variables. In this case, we will isolate y.

-4 = 2x – 1y

subtract 2x from both sides

-4 – 2x = 2x – 2x -1y
-4 – 2x = -1y
divide both sides by -1
-4/-1 – 2x/-1 = -1y/-1
4 + 2x = y

2. Plug the result from step one into the second equation. Afterwards, The second equation should all be in terms of one variable.

-2 = 3x – 1y
-2 = 3x -1(4 + 2x)

distributing the -1 into the 4 and 2x,

-2 = 3x – 4 – 2x

combining like terms,

-2 = x – 4

3. Solve the second equation for that variable.

-2 = x – 4

add 4 to both sides

-2 + 4 = x – 4 + 4

2 = x

4. Plug the result from step 3 into the first equation

-4 = 2x – 1y
-4 = 2(2) – 1y
-4 = 4 – 1y

subtract 4 from both sides

-4 – 4 = 4 -4 -1y
-8 = -1y

divide both sides by -1

-8/-1 = -1y/-1

8 = y

Hence we have found out both the values of x and y and solved the system of equations = (2,8)

Graphical Method

1. Change each equation into the slope – intercept form: y = mx + b. Essentially, y should be isolated on one side, and every other term should be on the other side.

Equation 1: -4 = 2x – 1y

subtract 2x from both sides

-4 – 2x = 2x – 2x – 1y
-4 – 2x = -1y

To isolate y, divide each term by -1

-4/-1 – 2x/-1 = -1y/-1
4 + 2x = y

  • slope = m = 2
  • y-int = b = 4

Equation 2: -2 = 3x – 1y

Subtract 3x from both sides

– 2 – 3x = 3x – 3x -1y
-2 – 3x = -1y

To isolate y, divide each term by -1

-2/-1 – 3x/-1 = -1y/-1
2 + 3x = y

  • slope = m = 3
  • y-int = b = 2

2. Graph each line on the standard coordinate plane.

Two Variables

3. Locate point of intersection. That is the solution to the system.

Two Equations

Using the graphical method, you will get the same result as you can see both these equations intersect exactly at (2,8).

There are three possible results when you work with these equations which includes:

  • one solution
  • no solution
  • infinite solutions

One Solution: If the system of equation contains two variables and has only one solution, it is called as ordered pairs and it means the solution arrived is for both the equations.

No Solution: If you have 2 lines that run parallel next to each other, then is never a possibility for both lines to intersect. This means the lines don’t have any points in common and hence you will not get a result.

Infinite Solutions: If you have 2 lines lying on top of the other, then there is a possibility of infinite number of solutions. Hence, a solution that works for one equation will also apply for the other equation.