Simultaneous equations can be daunting at first, but with the right approach and practice, you can learn how to solve them quickly and effectively. In this guide, we’ll explain what simultaneous equations are, some tips for solving them, and provide a few examples of how to work through them.
Simultaneous equations are two or more mathematical equations involving the same variables that must be solved together. This means that if you know the value of one variable in one equation, then it will also apply to the other equation. For example, if you have two equations: x + y = 10 and x - y = 2, then whatever value you assign to either ‘x’ or ‘y’ will affect both equations.
The key to solving simultaneous equations is identifying which method works best for the problem at hand. There are three main methods: substitution, elimination, and graphing.
Substitution involves assigning a value to one variable from one of the equations and using it to solve for another variable in the other equation. This is often easier for novice mathematicians as it only requires simple substitutions within each equation.
For example:
Solve for x and y in the following simultaneous equations:
2x + 3y = 12
4x – 5y = -6
Step 1: Assign x a value from either equation (we will use 2x + 3y = 12). Substitute 4x – 5y with 2(2x + 3y) so that 4x – 5y becomes 8x + 15y = 24.
Step 2: Subtract 8x from both sides of 8x + 15y = 24 giving us 15y = 16.
Step 3: Divide both sides by 15 giving us y = 1.Step 4: Now that we have found our value for y (1), we can substitute this back into either equation to find our value for x (in this case we will use 2x + 3(1) = 12). Doing this gives us 2x = 9 meaning x=4½ . Hence our final answer is x=4½, y=1.
Elimination also involves substitution but instead of assigning a variable a value from one equation; values from each side of an equation are added together so that either ‘x’ or ‘y’ is eliminated from both equations altogether. This method often works best when dealing with large numbers or long equations as it simplifies things considerably.
For example: Solve for x and y in the following simultaneous equations:2x - y = 8
3x - 7y = 6
Step 1: Add 7 times the first equation (7(2x - y) )to second equation so it becomes 11(2) – 7(-1)= 16+7= 23 giving us 11=23
Step 2: Divide each side by 11 giving us 1=2 Step 3: We can now substitute this into either original equation (we will choose 3(1) – 7(- 1)) giving us 3-7=- 4 meaning x=- 4/3 . Now we can substitute this back into either original equation (we will choose 2(- 4/3) - (- 1)) giving us -8/3+1=- 5/3 meaning y=- 5/3 . Hence our final answer is x=- 4/3 , y=- 5/3.
Graphing involves plotting each point on an XY graph before connecting them up with a straight line and finding where they intersect (the point at which both lines cross over). This method may seem intimidating at first but once you get comfortable with plotting points on a graph it actually becomes quite straightforward! Many modern calculators feature graphing functions so if you don't feel comfortable doing it by hand there's no need to worry!
For example : Solve for x and y in the following simultaneous equations:8-2y=6
6+5X=9
Step 1 : Plot both points on an XY graph; Point A is on X axis at 6 & Y axis at 0 , Point B is on X axis at 9 & Y axis at -2
Step 2 : Connect Point A & B using straight line
Step 3 : The point where these lines intersect is your answer ; hence our final answer is x=1 , Y=-2
It may seem daunting when confronted with multiple simultaneous equations but here are some tips to help make things easier : • Always double check your calculations as mistakes can easily be made when dealing with multiple variables! If necessary draw out