solve rs=3y+1,st=2y,rt=21
no unique solution, since you have 4 variables but only 3 equations.
To find some solutions, you could do this:
divide the 3rd by the 2nd
rt/(st) = 21/(2y)
r = 21s/(2y)
sub into the 1st
s(21s)/(2y) = 3y+1
21s^2 = 2y(3y+1)
s^2 = 2y(3y+1)/21
s =√[2y(3y+1)/21 ]
also t = 2y/s and r = 21/t
so you could assign any value to y and get the others
e.g. let y = 1
then s = √(8/21) = appr 6.172
then t = 3.24 and r = 6.48
so (r,s,t,y) = (6.48, .6172, 3.24, 1)
let y = 4
s = √(104/21) = appr 2.225
then t = 3.595 and r = 5.842
(r,s,t,y) = (5.842, 2.225, 3.595, 4)
I checked the above, if you store the 4 numbers in 4 separate memory locations on your calculator, the results are correct.
Without a calculator the above would be a nightmare
To solve this system of equations, we can use the method of substitution:
Step 1: Start with the equation rs = 3y + 1.
Step 2: Replace the "s" in the equation with its value from the equation st = 2y. Since st = 2y, we can substitute "st" in place of "s" in the first equation: rt = 2y * r = 3y + 1.
Step 3: Simplify the equation: 2ry = 3y + 1.
Step 4: We can further simplify by dividing both sides of the equation by y, yielding 2r = 3 + 1/y.
Step 5: Next, we solve the second equation st = 2y for y. By substituting the value of "st" from the third equation rt = 21: 2y = 21.
Step 6: Simplify the equation: y = 21/2 = 10.5.
Step 7: Substitute the value of y into the expression we found in Step 4: 2r = 3 + 1/10.5.
Step 8: Simplify the equation further: 2r = 3 + 2/21.
Step 9: Combine like terms and solve for r: 2r = 3 + 2/21. Multiplying both sides by 21 to clear the fraction gives us: 42r = 63 + 2.
Step 10: Simplify further: 42r = 65.
Step 11: Divide both sides by 42 to solve for r: r = 65/42 = 1.55 (rounded to two decimal places).
So, the solutions to the given system of equations are r = 1.55 and y = 10.5.