Simultaneous Equations

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Description of Levels

Projectable - A set of simultaneous equations designed to be shown one at a time to the whole class.

Level 1 - Equations that can be added or subtracted to eliminate one variable.

Level 2 - Equations that can be added or subtracted to eliminate one variable after one of the equations has been multiplied by a constant.

Level 3 - Equations that can be added or subtracted to eliminate one variable after both of the equations have been multiplied by constants.

Level 4 - Equations with two variables that are not written in the standard way.

Level 5 - Real life problems that can be solved by writing them as simultaneous equations.

Level 6 - Equations which include fractions in some way.

Level 7 - Linear, quadratic and other pairs of simultaneous equations.

These Level 7 questions will require you to be able to solve Quadratic Equations.

Exam Style Questions - A collection of problems in the style of GCSE or IB/A-level exam paper questions (worked solutions are available for Transum subscribers).

More Simultaneous Equations including lesson Starters, visual aids, investigations and self-marking exercises.

Extension

There is a printable worksheet to go with this activity. It is an exercise that appeared in an algebra book published in 1895. It starts with basic questions but soon gets tricky!

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Curriculum Reference

See the National Curriculum page for links to related online activities and resources.

Help Video

The examples used in the video are available to teachers as projectable slides.

Level 7 Example

\(x-y=1 \qquad \mathbf{A}\\y=x^2-21 \qquad \mathbf{B}\)

One way of solving this pair of simultaneous equations is to use the substitution method.
Equation \(\mathbf{A}\) can be rearranged to make \(x\) the subject

\(x = 1 + y\)

This expression for \(x\) can be substituted into equation \(\mathbf{B}\):

\(y=(1+y)^2-21\\y=1+2y+y^2-21\\y=y^2+2y-20\\y^2+y-20=0\)

This is a quadratic equation that can be solved by factorisation.

\( (y-4)(y+5) = 0\)

Either \(y=4\) in which case, by substituting into equation \(\mathbf{A}\), \(x=5\)

Or \(y=-5\) in which case, by substituting into equation \(\mathbf{A}\), \(x=-4\)

These Level 7 questions will require you to be able to solve Quadratic Equations.

Alternate method

\(x-y=1\\y=x^2-21\)

Get both equations into the "y=" format

\(y=x-1 \\y=x^2-21\)

Set the equations equal to each other

\(x^2-21 = x-1\)

Rearrange into the standard quadratic equation format

\(x^2-x-20 = 0\)

Solve the quadratic equation

\((x-5)(x+4) = 0\)

\(x=5 \qquad or \qquad x=-4\)

Use the linear equation to get matching y values

When \(x = 5, \qquad y = 4\)
When \(x = -4, \qquad y = -5\)

You can check your answers by substituting them both into equation \( \mathbf{B} \) to see if it balances.

This example is not intended to teach you everything you need to know about this type of simultaneous equations. It is here as a reminder and is no substitute for your teacher or tutor.