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A system of equations is a group of one or more equations with several variables. To solve a system, you must identify all of the common solutions or points of intersection.

We require at least 2 equations to solve a system of equations in 2 variables. Similarly, we will need at least 3 equations to solve a system of equations in 3 variables.

3 ways to solve a system of equations:

Substitution Method:

express y in terms of x in one of the equations and then substitute it in the 2nd equation.

Elimination Method:

  • Step 1: Eliminate one of the unknowns, by multiplying equations by suitable numbers, so that the coefficients of one of the variables become the same.
  • Step 2: Subtract one equation from the other to find the value of one of the unknowns.
  • Step 3: Substitute the value of the found unknown into one of the original equations to calculate the second unknown value.
  • Step 4: To verify the correctness of the answer, take both values and substitute them in an equation that was not used to find the value of the second unknown.

Graphical Method:

plot the graphs of simultaneous equations to get the solution.

There can be different types of solutions to a given system of equations:

  • Unique solution
  • No solution
  • Infinitely many solutions

Example 1:

x = 4 + 2y

2x - 4y = 5

2(4 + 2y) - 4y = 5

8 + 4y - 4y = 5

8 = 5

Answer: The given system of equations has no solution.

Example 2:

Solution of a system of equations with two variables by the graphical method.

Solve and graph the system below:

x + 3y = 7

y = -2x - 1

Solving system of equations


If there are two unknowns:

Both equations are taken and brought to the "y =" state so that we could draw a graph on the coordinate line.

The intersection points of these two graphs will be the answers.

Solution Example 2

Example 3:

If the system has exactly one solution, then k = ( ) ?

y = x^2- x + 9

y = kx


Solution Example 3 part 1

To have exactly one solution, the discriminant = 0

Solution Example 3 part 2

Answer: k = -7 or k = 5