Solve the simultaneous equations 7x - (3sqrt(5)) * y = 9sqrt(5) and
(2sqrt(5)) * x + y = 34
AAAaannndd the bot gets it wrong yet again!
7x - 3√5 y = 9√5
2√5 x + y = 34
x = 3√5
y = 4
Apologies for the incorrect response. Let's solve the simultaneous equations correctly:
Equation 1: 7x - 3√5y = 9√5
Equation 2: 2√5x + y = 34
We will solve this system of equations using the method of substitution:
Step 1: Solve Equation 2 for y:
y = 34 - 2√5x
Step 2: Substitute the value of y in Equation 1:
7x - 3√5(34 - 2√5x) = 9√5
Step 3: Simplify the equation:
7x - 3√5(34) + 3√5(2√5x) = 9√5
7x - 102√5 + 30√5x = 9√5
Step 4: Combine like terms:
7x + 30√5x - 102√5 = 9√5
Step 5: Combine like terms:
37x - 102√5 = 9√5
Step 6: Add 102√5 to both sides:
37x = 18√5 + 102√5
Step 7: Combine like terms:
37x = 120√5
Step 8: Divide both sides by 37:
x = (120√5)/37
Step 9: Simplify the expression:
x = (24√5)/37
Now, substitute the value of x back into Equation 2 to find y:
2√5((24√5)/37) + y = 34
Simplify:
(48√5/37) + y = 34
Step 10: Subtract (48√5/37) from both sides:
y = 34 - (48√5/37)
y = (34*37 - 48√5)/37
Therefore, the solution to the simultaneous equations is x = (24√5)/37 and y = (34*37 - 48√5)/37.
To solve the simultaneous equations:
Step 1: Let's eliminate one variable by multiplying the first equation by (2√5) and the second equation by (3√5):
(2√5) * (7x - (3√5) * y) = (2√5) * (9√5)
(3√5) * ((2√5) * x + y) = (3√5) * 34
Simplifying these equations, we get:
14√5x - 30y = 90
6√5x + 3√5y = 102
Step 2: Now, add the two equations together to eliminate x:
(14√5x - 30y) + (6√5x + 3√5y) = 90 + 102
Simplifying the equation, we get:
20√5x - 27√5y = 192
Step 3: Divide the equation by 3 to simplify it further:
(20√5x - 27√5y)/3 = 192/3
Simplifying this equation, we get:
20/3√5x - 9√5y/3 = 64
Step 4: Now, let's rearrange the equation to isolate one variable. Let's isolate x:
20/3√5x = 9√5y/3 + 64
Simplifying the equation, we get:
x = (9√5y/3 + 64) / (20/3√5)
Step 5: Simplify the expression within the parentheses:
x = (9√5y/3 + 64) / (20/3√5)
x = (3√5y + 192) / (20√5)
Step 6: Substitute this expression for x in either of the original equations. Let's use the first equation:
7x - (3√5)y = 9√5
7((3√5y + 192) / (20√5)) - (3√5)y = 9√5
Step 7: Simplify the equation:
7(3√5y + 192) / (20√5) - (3√5)y = 9√5
Step 8: To clear the fraction, we can multiply the entire equation by (20√5):
7(3√5y + 192) - (20√5)(3√5)y = (20√5)(9√5)
Step 9: Simplify the equation:
21√5y + 1344 - 300y = 900
Step 10: Combine like terms:
-279√5y + 1344 = 900
Step 11: Solve for y:
-279√5y = 900 - 1344
-279√5y = -444
Step 12: Divide both sides of the equation by -279√5 to isolate y:
y = -444 / -279√5
Step 13: Simplify the expression:
y = 4√5 / 3
Step 14: Now substitute the value of y back into one of the original equations. Let's use the second equation:
(2√5)x + (4√5 / 3) = 34
Step 15: Solve for x:
(2√5)x = 34 - (4√5 / 3)
Step 16: Simplify the expression:
(2√5)x = (102 - 4√5) / 3
Step 17: Divide both sides of the equation by 2√5 to isolate x:
x = (102 - 4√5) / (6√5)
Step 18: Simplify the expression:
x = (17 - (2/3)√5) / (√5)
Therefore, the solutions to the simultaneous equations are:
x = (17 - (2/3)√5) / (√5)
y = 4√5 / 3
To solve the simultaneous equations:
Equation 1: 7x - (3√5)y = 9√5
Equation 2: (2√5)x + y = 34
We can solve this system of equations using the method of substitution.
Step 1: Solve Equation 2 for y:
y = 34 - (2√5)x
Step 2: Substitute the value of y in Equation 1:
7x - (3√5)(34 - (2√5)x) = 9√5
7x - 102√5 + (6√5)x = 9√5
Step 3: Simplify the equation:
7x + (6√5)x - 102√5 = 9√5
Step 4: Combine like terms:
13x - 102√5 = 9√5
Step 5: Add 102√5 to both sides:
13x = 18√5 + 102√5
Step 6: Combine like terms:
13x = 120√5
Step 7: Divide both sides by 13:
x = (120√5) / 13
Step 8: Simplify the expression:
x = (24√5) / 13
Now, substitute the value of x back into Equation 2 to find y:
(2√5)(24√5/13) + y = 34
Simplify:
(48/13)(√5)^2 + y = 34
(48/13)(5) + y = 34
(240/13) + y = 34
y = 34 - (240/13)
y = 442/13 - 240/13
y = 202/13
Therefore, the solution to the simultaneous equations is x = (24√5)/13 and y = 202/13.