Solve the following pair of simultaneous equations: 2y - x = 3, y + 4x = 4

There are 2 main methods for solving simultaneous equations: 1) by substitution, and 2) by elimination. Have a look at how each method works and pick the one that's best for you! In order to make my explanations easier to follow, I will label the equation 2y - x = 3 as A) and y = 4x + 4 as B)

Make sure you have a go at each method and try your best to understand both, as sometimes textbooks/resources will only show working using a specific method.

Method 1) Substitution

Step 1: Rearrange to make either x or.y the subject, whichever is easier.E.g Rearranging A) to make x the subject gives x = 2y - 3

Step 2: Substitute this into the second equation.Substituting x = 2y - 3 into B) gives y = 4(2y - 3) + 4

Step 3: Rearrange and simplify.Need to get all the y's on one side of the equals sign! Rearranging y = 4(2y - 3) + 4 gives 8 = 7y. Hence dividing both sides by 7, gives y = 8/7

Step 4: Don't forget to find x too!(You can use either equation A) or B) to find x; I decided to use A)) 2(8/7) - x = 3. Rearranging for x gives us x = -5/7

Step 5: Check your answer! Substitute your answer into A) or B) and see if it works. If it doesn't, you need to go back and check your working - it's usually some dodgy arithmetic that can let you down!


Method 2: EliminationThe aim is to add or subtract a multiple of A to/from B to get an equation with just y's (so eliminate the x's)or just x's (eliminate the y's)To make this easier it's best for both equations to be rewritten in the following form: A) 2y - x = 3 and B) y - 4x = 4

Step 1: Have a look at the coefficients (numbers in front of) the x's and y's. Here I chose to eliminate the y's because it involves some slightly less complicated arithmetic! Multiplying B) by 2 (remember to multiply every single term of the equation, on both side of the = sign) gives you 2y - 8x = 8

Step 2: Do the adding or subtracting bit.Now we have A) 2y - x = 3 and B) 2y - 8x = 8. Both have 2y in them, so we need to subtract one equation from the other. (If one equation had 2y and the other -2y then we'd need to add instead). So A) - B) gives us 7x = - 5. Dividing both sides by 7, gives us x = -5/7

Step 3: Don't forget to find y too! Substitute you value for x into either A) or B) and rearrange and simplify. This gives us y = 8/7

Step 4: Check your working! Substitute your answer into A) or B) and see if it works.