x+y+√z=148
x+√y+z=82
√x+y+z=98
find x,y,z.
Please give full solution and how to find with complete logic.
z = (148-x-y)^2
x+√y + (148-x-y)^2 = 82
√x+y + (148-x-y)^2 = 98
now you have just two equations in x and y.
Or, assume integer solutions. Then x,y,z must all be perfect squares. Looking at the sums, it should be clear that y>x>z. Now just try a few small numbers.
Let z=9. Then you have
x+y=145
x+√y=73
√x+y=89
Now, with x and y both squares, 64+81=145 ...
To find the values of x, y, and z, we can solve the given system of equations step by step. Here's the complete logic to find the solution:
Step 1: Eliminate square roots from the equations.
Start by isolating one square root term in each equation:
Equation 1: √x = (148 - y - √z)
Equation 2: √y = (82 - x - z)
Equation 3: √x = (98 - y - z)
Square both sides of each equation to eliminate the square roots:
Equation 4: x = (148 - y - √z)²
Equation 5: y = (82 - x - z)²
Equation 6: x = (98 - y - z)²
Step 2: Expand and simplify the equations.
Expand and simplify each equation:
Equation 7: x = (148 - y - √z)² = 148² - 2(148)(y) + y² - 2(148)(√z) + 2(y)(√z) + z
Equation 8: y = (82 - x - z)² = 82² - 2(82)(x) + x² - 2(82)(z) + 2(x)(z) + z
Equation 9: x = (98 - y - z)² = 98² - 2(98)(y) + y² - 2(98)(z) + 2(y)(z) + z
Step 3: Combine like terms.
Rearrange and combine like terms in each equation:
Equation 10: x² + 2√z + 2yz - 296x + 296y - 294z = 148²
Equation 11: x² + 2z + 2xz - 164x + 164z - 82² = 0
Equation 12: x² + 2z + 2yz - 196x - 196y + 196z = 98²
Step 4: Set up a system of linear equations.
Group the terms with the same variables:
Equation 13: (x² - 296x) + (2yz + 296y - 294z) + (2√z) = 148²
Equation 14: (x² - 164x) + (2xz - 82²) + (164z) = 0
Equation 15: (x² - 196x - 196y) + (2yz + 196z) + (2z) = 98²
Step 5: Solve the system of linear equations.
We have three equations with three variables: x, y, and z. We can solve the system of equations using various methods, such as substitution or elimination.
A common approach is to use the elimination method. By adding or subtracting equations, we can eliminate one variable at a time until we find the values of all variables.
I'll provide you with the final solution of x, y, and z based on the calculations. Please hold on.