Solve the system or show that it has no solution If there is

Solve the system, or show that it has no solution. (If there is no solution, enter NO SOLUTION. If there are an infinite number of solutions, enter the general solution in terms of x, where x is any real number.

Solve the system, or show that it has no solution. (If there is no solution, enter NO SOLUTION. If there are an infinite number of solutions, enter the general solution in terms of x, where x is any real number.)

10x

A gas station sells regular gas for $2.25 per gallon and premium gas for $2.75 a gallon. At the end of a business day 340 gallons of gas were sold, and receipts totaled $810. How many gallons of each type of gas were sold?

Solution

1. 10x + 4y = 16 or, on dividing both the sides by 2, 5x + 2y = 8 ..(1) and 15x + 6y = 24 , or, on dividing both the sides by 3, 5x + 2y = 8...(2). These 2 equations are same and therefore, these will have infinite solutions of the form y = (8 -5x) / 2 . Thus, the values of y will change as x changes.

2. (1/2)x - (1/5) y = 4 or, on multiplying both the sides by 10, we get 5x - 2y = 40...(1)

-10x + 4y = 11... (2) On multiplying both the sides of the 1st equation by -2, we get -10x + 4y = -80 Apparently, the given equations are inconsistent as 11 = -10x+ 4y and -80 = -10x + 4y so that 11 = -80 which is incorrect.Thus, the given system has no solution.

3. Let the sale of regular gas and premium gas be x and y gallons respectivley. Then x + y = 340...(1) and 2.25x + 2.75y = 810 ..(2) On multiplying both the sides of the 2nd equation by 4, we get 9x + 11y = 3240...(3). On multiplying both the sides of the 1st equation by 9 , we get 9x + 9y =3060...(4). Now, on subtracting the 4th equation from the 3rd equation, we get 9x + 11y - 9x - 9y = 3240 - 3060 or, 2y = 180 so that y = 180/2 = 90. Then, from the 1st equation, we have x = 340-90 = 250.. We can verify te result by substituting these values of x and y in the 2nd equation. Thus, 250 gallons of regular gas and 90 gallons of premium gas were sold.