Simultaneous equations

To solve two sets of equations algebraically at the same time:

1. Number the equations (1) and (2)
2. Look to see what needs to be done to lose one of the variables:-
   • Substitute one equation for another
   •  Add both equations
   • Subtract both equations
   • Multiply one equation, then add.
   • Multiply both equations then add.
3. Solve to find one variable (eg x or y)
4. Substitute that value into equation (1).
5. Solve to find other variable.
6. Check by substituting both values into equation (2)
7. Write down solution.

 

Elimination by Substitution   

Here, we are looking to eliminate a variable by substituting one equation into the other.

 

Example

Number the equations (1) and (2)

Substitute equation (1) into (2)

Solve for x.

Now substitute the value of x back into one of the equations (your choice) and solve for y.

To check your answer, substitute your values for x and y into the equation that you did not use.

Elimination by Addition  

Here, we are looking to eliminate a variable by adding both equations.

Example

Notice that adding the equations will lose the y variable.

Number the equations (1) and (2), then add both equations:

Now substitute the value of x back into one of the equations (your choice) and solve for y.

 

 

Elimination by Subtraction

Here, we are looking to eliminate a variable by subtracting one equation from the other.

Example

Notice that subtracting the equations will lose the x variable.

Number the equations (1) and (2), then subtract:

Now substitute the value of y back into one of the equations (your choice) and solve for x.

           

Elimination by Multiplying one equation.

Here, we are looking to eliminate a variable by first multiplying one equation and then adding or subtracting the equations.

Example

Notice that doubling the bottom equation followed by adding the equations will lose the y variable.

Number the equations (1) and (2), then multiply the bottom equation by 2.

This gives a new equation, which can be added to the top equation:

Now substitute the value of x back into one of the original equations (your choice) and solve for y.

Elimination by Multiplying both equations.

Here, we are looking to eliminate a variable by first multiplying both equations and then adding or subtracting the equations.

Example

Number the equations (1) and (2), then multiply the bottom equation by -3 and the top equation by 2:

This gives two new equations, which can be added to lose the y variable:

Now substitute the value of x back into one of the original equations (your choice) and solve for y.

For solving graphically, click here