# Simultaneous Systems of Equations Solved by Algebraic Elimination

This applet investigates simple linear simultaneous equations (systems of equations) in two variables and how to solve them by eliminating one of the variables. You generate any number of random examples at six different levels of difficulty.

###### Author and programmer: Ron Barrow

UK Years 9-11, Higher GCSE Mathematics - Algebra

US Years 9-11, Grades 8 - 10 - Algebra 1

Instructions below See also: Simultaneous equations solved graphically

## How to Use this Applet

This program aims to help you understand and solve simple simultaneous equations (systems of equations) using a process called **elimination**.

The two equations to solve are at the top of the screen, marked as equations 1 (#1) and 2 (#2). The working will be seen below. Initially the two equations (#1 and #2) are copied as equations #3 and #4.

At the bottom of the screen you will see three buttons:

"#3 + #4" - adds the two equations #3 and #4 together

"#3 - #4" - subtracts bottom from top (4# from #3)

"#4 - #3" - subtracts top from bottom (3# from #4)

Try clicking these buttons to see what happens.

You can also multiply either of the two equations by a number. This is done using the drag-boxes beside them. Just place your cursor in the middle of the box, then click-and-drag, left or right, to change the number. That equation is then multiplied through by the number. Try it and see.

By combining these actions - multiplying equations through and then adding or subtracting them - you are aiming to **eliminate** one of the variables. That is, you're trying to get rid of either *x* or `y` in the third line (equation #5).

When you have eliminated either x or y, you have a much simpler equation to solve, equation 6 (#6). You solve this equation by dividing through by a number using the third drag-box which you should now see. If you divide equation #6 by the correct number you will see the rest of the calculations done for you. Otherwise try dividing again.

The working seen on the right finds the value of the variable that you eliminated before, using the one of the original equations #1 or #2. You can choose which of these two equations you want to use from the buttons at the bottom. It doesn't matter which you choose, as you'll get the same answer either way.

Solving simultaneous equations, or systems of equations, in this way is an important skill. But it is not easy at first. If you're going wrong somewhere, you can always press the "Start again" button, or the "New problem" button to get a new pair of equations to solve.

There are six levels of difficulty, but I recommend starting with level 1, where you probably will be able to progress by simply adding the two equations together.

Best of luck!