Find two consecutive even integers such that six times the first is three times the second. (Enter your answers as a comma-separated list.)

Answer 1

Since they must be even, the first term must be 2x since it is unknown (hence the variable x) but must also be even (which is why it is multiplied by 2. If 2x is the first term, the second term must be 2x+2 since it is the next even number which means it is the first term +2. From the phrase "six times the first is three times the second" we can create the equation 6(first term) = 3(second term). From this equation we can replace "first term" and "second term" with the variables we have made previously giving us 6(2x) = 3(2x+2). Multiplying out the 6 and the 3 will give us 12x = 6x+6. From here, solving for x gives us x = 1. But we're not done yet. The question asks us for the integers which means we need to take this value for x and plug it back into our expressions for the integers. For the first term replacing the x in 2x for 1 gives us 2(1) = 2. Meanwhile, for the second term, replacing the x in 2x+2 gives us 2(1)+2 = 4. Therefore the answers are 2 and 4, or in the form of a comma-separated list: 2, 4.