Given that seven 𝑥 divided by 𝑥 minus three is equal to 16𝑥 divided by 𝑥 plus three minus nine, find the value of 𝑥.
As the common denominator is 𝑥 minus three multiplied by 𝑥 plus three, we need to multiply all three terms in the equation by this common denominator. At this stage, we can cancel anything that is on the top and the bottom of each individual term. This leaves us with seven 𝑥 multiplied by 𝑥 plus three is equal to 16𝑥 multiplied by 𝑥 minus three minus nine multiplied by 𝑥 minus three multiplied by 𝑥 plus three.
Expanding the first term, seven 𝑥 multiplied by 𝑥 plus three, gives us seven 𝑥 squared plus 21𝑥. Expanding the second term, 16𝑥 multiplied by 𝑥 minus three is equal to 16𝑥 squared minus 48𝑥. And finally expanding the third term, negative nine multiplied by 𝑥 minus three multiplied by 𝑥 plus three, gives us negative nine 𝑥 squared minus 27𝑥 plus 27𝑥 plus 81, which can be simplified to negative nine 𝑥 squared plus 81 as negative 27𝑥 plus 27𝑥 equals zero.
Putting each of these expansions back into the equation gives us seven 𝑥 squared plus 21𝑥 equals 16𝑥 squared minus 48𝑥 minus nine 𝑥 squared plus 81. Collecting the like terms on the right-hand side, 16𝑥 squared minus nine 𝑥 squared gives us seven 𝑥 squared. And then subtracting seven 𝑥 squared from both sides of the equation gives us 21𝑥 equals negative 48𝑥 plus 81. Adding 48𝑥 to both sides of this equation leaves us 69𝑥 equals 81.
And finally, dividing both sides by 69 gives us a value of 𝑥 of 81 over 69. As both the numerator and denominator of this fraction are divisible by three, we can simplify the fraction so that our value of 𝑥 is 27 over 23 or 27 divided by 23. This means that despite the complexity of the initial equation, there is only one solution. 𝑥 equals 27 over 23 solves the equation seven 𝑥 divided by 𝑥 minus three is equal to 16𝑥 divided by 𝑥 plus three take away nine.