Solve the simultaneous equations using the substitution method. 2y+x=8 and 1+y=2x.

First point out that it makes sense to solve this system of equations using the substitution method because one of the variables can easily be isolated without multiplication or division.(Step 1:) Label the first equation (1) and the second equation (2) as this makes it easier to keep track of the method.(Step 2:) To use the substitution method, one of the variables must be isolated, we could use the 'x' in (1), but on this occasion will choose the 'y' in (2). This is isolated by subtracting 1 from both sides, giving y=2x-1.(Step 3:) It is important to understand that the unknowns in the equations, y and x, both represent numbers and are consistent throughout both equations, meaning whatever value x takes in the first equation it also takes in the second equation, and the same is true for y. Using this fact, we can substitute the expression which y is given as in the second equation into the first equation, as y is the same in both equations. So we replace the y in equation 1 with 2x-1 to give 2(2x-1)+x=8. This has given us an equation with only one unknown, x, and so we can solve it using algebra. (Step 4:) To solve this, we first expand the bracket, by multiplying the terms inside by the 2 on the outside to give 4x-2+x=8.(Step 5:) We can now collect the like terms (i.e all terms including x) to get 5x-2=8.(Step 6:) Add 2 to both sides, giving 5x=10, and finally divide both sides by 5 to give x=10/5=2.(Step 7:) We now need to find y using our value of x, which we can do by substituting this back into our second equation, which y is already the subject of (so there is no need to do any more algebra). This gives 2(2)-1=4-1=3. The solution is therefore x=2 and y=3. (Step 8:) It is always good practice to check that this solution works by substituting them into the other equation and checking that it gives the expected answer. So letting x=2 and y=3, equation 1 becomes 2(3)+(2)=8, which is true, so the solution is correct.