# Simultaneous Equations

## A self-marking, multi-level set of exercises on solving pairs of simultaneous equations.

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This is level 5: real life problems that can be solved by writing them as simultaneous equations. You will be awarded a trophy if you get at least 9 correct and you do this activity online.

 What two numbers have a sum of 21 and the smaller subtracted from the larger equals 3? smaller number= larger number= What two numbers add up to 43 and have a difference of 15? smaller number= larger number= What two numbers add up to -2 and have a difference of 10? smaller number= larger number= There are some rabbits and chickens in a field. Altogether there are 75 heads and 222 legs. How many of each were there? numbers of rabbits= and chickens= The perimeter of a rectangle is 126cm and its length is 33cm longer than its width. Find, in centimetres, the dimensions of the rectangle. rectangle length= width= Find two numbers such that twice the first added to three times the second equals 203 and the first added to twice the second equals 128. the first number= second number= The difference between the size of the two acute angles of a right-angled triangle is 20o. Find the size of these two acute angles in degrees. the smaller angle= larger angle= Six years ago a father was twelve times as old as his son. In three years time the father will be three times as old as his son. How old are they now? the father’s age= son’s age= Three years ago a mother was four times as old as her daughter. In nine years time the mother will be twice as old as her daughter. How old are they now? the daughter’s age= mother’s age= 11 nuts and 14 bolts weigh 434g together while 7 nuts and 28 bolts weigh 658g. What is the weight of a single nut and a single bolt in grammes? weight of one nut= one bolt= Last month Anthony bought 20 cups of coffee and 26 cups of tea from the studio canteen. They cost him £32.68 altogether. Declan bought 14 cups of coffee and 41 cups of tea and they cost him £40.66 altogether. How much do cups of tea and coffee cost in pence? one cup of coffee= cup of tea= The perimeter of a rectangle is 104cm and its area is 420cm2. Given that the length is longer than the width find, in centimetres, the dimensions of the rectangle. rectangle length= width=
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This is Simultaneous Equations level 5. You can also try:
Level 1 Level 2 Level 3 Level 4 Level 6 Level 7

## Instructions

Try your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help.

When you have got all of the questions correct you may want to print out this page and paste it into your exercise book. If you keep your work in an ePortfolio you could take a screen shot of your answers and paste that into your Maths file.

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## Description of Levels

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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!

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

## 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 5 Example

What two numbers add up to 50 and have a difference of 12?

Let the smaller number be $$x$$ and the larger number be $$y$$.

$$y+x=50 \qquad \mathbf{A}\\y-x=12 \qquad \mathbf{B}$$

Add equation $$\mathbf{A}$$ to equation $$\mathbf{B}$$

$$2y=62$$
$$y=31$$

Substitute this value for $$y$$ into equation $$\mathbf{A}$$.

$$31+x=50$$
$$x=19$$

The simultaneous equations have been solved.
The smaller number is 19 and the larger number is 31.

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.

## Simultaneous Equations Checklist

1. Decide if the equations are in the correct form.
2. Decide if we need to manipulate one or both equations.
3. Decide if we need to add or subtract.
4. Successfully add or subtract algebraic expressions, possibly involving negative numbers.
5. Solve a linear equation, possibly involving negative numbers.
6. Substitute the solution into an algebraic expression.
7. Solve another linear equation.
8. Solve another linear equation.
9. Substitute two solutions into an algebraic expression to check the answer.
10. Interpret the solution.

These steps are developed and discussed in "How I Wish I'd Taught Maths: Lessons learned from research, conversations with experts, and 12 years of mistakes" by Craig Barton

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can double-click the 'Check' button to make it float at the bottom of your screen.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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