Final answer:
To prove that 3x + 8y is rational if x and y are rational, we substitute x and y with their respective rational expressions and simplify the equation to show that the result is a quotient of two integers, which proves it is rational.
Step-by-step explanation:
To prove that 3x + 8y is rational if x and y are rational, we need to show that the sum of two rational numbers is also rational.
Let's assume x and y are rational numbers. This means they can be expressed as a quotient of two integers, where the denominator is not equal to zero.Let's say x = a/b and y = c/d, where a, b, c, and d are integers and b, d ≠0. Now we can substitute these values into the expression 3x + 8y:3x + 8y = 3(a/b) + 8(c/d) = (3a/b) + (8c/d) = (3a + 8c)/(b*d)The numerator, 3a + 8c, is the sum of two integers, and the denominator, b*d, is the product of two non-zero integers. Therefore, 3x + 8y is a quotient of two integers and is rational.