How do I solve a simultaneous equation like this: 2x-5y=3, 3x+2y=14 ?

Solving a simultaneous equation finds the point(s) where two lines on a graph intercept. Therefore, one way of solving the equations is to plot the two lines onto the same graph and find the coordinates of the point where they cross. This is a good method to visualize the question but in practice it’s better to use either the elimination or the substitution method. For the elimination method you need to eliminate either x or y from the equations. If you choose to eliminate y, you first need to make both y terms the same. This can be done by multiplying the first equation by 2 and the second equation by 5. Make sure to multiply all of the terms on both sides of the equal signs: 4x-10y=6, 15x+10y=70. Now you must eliminate y by either adding or subtracting the equations from each other. If the signs before the y terms are the same you subtract, if the signs are different you add. In this example there’s one + and one – y term so you add the two equations and can then find the value of x: 19x=76… x=4. Now you know x you must put it back into one of the equations and solve for y: 2(4)-5y=3… 8-3=5y… 5=5y… y=1. To make sure you have the right answer put your x and y values into the other equation (the one not used in the last step) and check that you get the correct answer.

Alternatively, you can use the substitution method. (This method must be used when the simultaneous equations include quadratics). For this method you must rearrange one equation to get x= or y=. Using the first equation you get x=1.5+2.5y. Substitute this in for x in the second equation: 3(1.5+2.5y)+2y=14. You can then solve for y: 4.5+7.5y+2y=14… 9.5y=9.5… y=1. Like in the elimination method you must then put this y value back into one of the original equations to find x.