Solve the simultaneous equations 3x + y = -4 and 3x - 4y = 6

When solving simultaneous equations our goal is always to use the equations given, which each have two unknowns in them, in order to find an equation with only one unknown. In this example, we can see that both equations have 3x on the left hand side, so we can subtract equation 2 from equation 1 to eliminate the x term, giving (3x+y) - (3x-4y) = -4 -6, so 5y = -10 and hence y = -2. From here we can use either equation to find x by substituting the value -2 for y. If we use equation 1, we see that 3x + y = 3x - 2 = -4, hence 3x = -2 and so x = -2/3. For ease of mind we can also check that this agrees with equation 2: 3(-2/3) -4(-2) = -2+8 = 6. We can now be sure that our unknowns x and y satisfy both equations and are -2/3 and -2 respectively.