Solving simultaneous equations algebraically

Algebraic method

You can solve simultaneous equations by adding or subtracting the two equations in order to end up with an equation with only one unknown value.

This is known as the algebraic method.

Example

Solve the simultaneous equations:

2x + y = 9(1)

3x - y = 1(2)

Add the two equations together and you will find that the y disappears:

3x + 2x + y - y = 9 + 1

This can be simplified to:

5x = 10

x = 2

Substituting this value of x in (1) gives:

4 + y = 9

y = 5

Check in (2):

6 - 5 = 1 (which is correct)

So the solution is:

x=2, y=5

Question

Solve the simultaneous equations

3x + 2y = 8

3x - y = 5

Firstly use labels for the equations:

3x + 2y = 8 (1)

3x - y = 5 (2)

To make the 3xs disappear we can subtract equation (2) from equation (1):

3x - 3x + 2y - (-y) = 8 - 5

This simplifies to:

3y = 3

y = 1

Substitute in (1):

3x + 2 = 8

3x = 6

x = 2

Check in (2):

6 - 1 = 5 (which is correct)

So the solution is:

x = 2, y = 1

Sometimes equations need to be altered, by multiplying throughout, before being able to eliminate one of the variables (letters).

Example

Solve the simultaneous equations:

{2x} + {3y} = {9} (1)

{3x} +  {y}  = {10} (2)

Neither the x nor the y will be eliminated by adding or subtracting these equations as they stand.

By multiplying the second equation by {3} throughout, both equations will then include {3y}, which will allow us to continue with the solution.

(2) \times~{3} gives {9x} + {3y} = {30} (3)

Don’t forget to multiply the right hand side by 3 as well.

Now, looking at equations (1) and (3), as they both include ‘ +{3}{y}’, we will subtract one equation from the other.

Subtract equation (1) from equation (3) and you’ll find that the y disappears:

(9x+3y)-(2x+3y)=30-9

This can be simplified to:

{7x} = {21}

{x} = {3}

Substituting this value of x in (1) gives:

{2}\times{3}+{3y}={9}

{6}+{3y}={9}

{3y}={3}

{y}={1}

Check in (2):

{9} + {1} = {10} (which is correct)

So the solution is:

{x}={3}, {y}={1}

Question

Solve the simultaneous equations:

{5x} + {3y} = {21}

{x} - {y} = {1}

Firstly use labels for the equations:

{5x} + {3y} = {21} (1)

{x} - {y} = {1} (2)

To make the ys disappear we need to multiply equation (2) by {3} before adding to equation (1):

(2) \times~{3} gives {3x} - {3y} = {3} (3)

By adding equation (1) and equation (3) and you will find that the {y} disappears:

({5x} + {3y}) + ({3x} - {3y}) = {21} + {3}

This can be simplified to:

{8x} = {24}

{x} = {3}

Substitute in (1):

{5}\times{3}+{3y}={21}

{15}+{3y}={21}

{3y}={6}

{y}={2}

Check in (2):

{3} - {2} = {1} (which is correct)

So the solution is:

{x}={3}, {y}={2}