Note: this page contains legacy resources that are no longer supported. You are free to continue using these materials but we can only support our current worksheets, available as part of our
membership offering. Simultaneous equations are two equations, each with the same two unknowns and are “simultaneous” because they are solved together. Click below for the Simultaneous Equations Worksheet Generator which provides limitless questions for practice. In simple terms, the solution to a pair of simultaneous equations
is the x and y values of the coordinates of the point at which the graphs cross or intersect. The example below shows this. For each equation, find coordinates for two points on the graph. An easy way of doing this is finding corresponding values when x = 0 and when y = 0. (Note: this will give 2 sets of coordinates which, since the equation is linear, is enough although it is a god idea to check at least one more
point on the line.) For 4x – 2y = 10 this gives (2.5, 0) and (0,-5) which we plot and then extend a straight line through.  For x + y = 4 this gives (0,4) and (4,0) which we plot and then extend a straight line through. Notice that the all the coordinates through which the lines pass are solutions to each equation. And the coordinates of the point at which they cross, (3,1)is the solution to the pair of simultaneous equations. Solving AlgebraicallyWe can find solutions to simultaneous equations algebraically too. There are two common methods. Which one you choose might depend on the values involved or it might just be the method you like the most. We will use the same pair of equations as above. Elimination Method
It is always a good idea to check the values for x and y in the other equation. Substitution Method
And, as a check, try the values for x and y in the other equation. More Examples Using Algebraic MethodsThere is one example of each the elimination and substitution methods for solving simultaneous equations shown below. Elimination MethodSubstitution MethodSimultaneous Equations in Real-lifeSam and Jack have $50 between them and Sam has $5 more than Jack. How much money does each have? s + j = 50 s = 27.50 , j = 22.50 This example is quite simple – you might be able to work it out by trial-and-error – but you can use any of the methods above to solve it. And finally,… Remember to practice with questions from the Simultaneous Equations Worksheet Generator. Link/Reference UsWe spend a lot of time researching and compiling the information on this site. If you find this useful in your research, please use the tool below to properly link to or reference Helping with Math as the source. We appreciate your support!
What is the solution of simultaneous equations?If you have two different equations with the same two unknowns in each, you can solve for both unknowns. There are three common methods for solving: addition/subtraction, substitution, and graphing. This method is also known as the elimination method.
What is the easiest method for simultaneous equations?The elimination method is the easiest method for solving the simultaneous equations because the y-terms have the same coefficient with opposite signs, so when adding these equations the y-terms cancel out immediately and then solving for the x-term is simple.
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