find the value of x,yandz in the system equation
2x+y-z=8 x^2-y^2+2z^2=14 3x^3+4y^3+z^3=195
step please thanks
2x+y-z=8
x^2-y^2+2z^2=14
3x^3+4y^3+z^3=195
See whether it helps to rewrite it like this
x+y + x-z = 8
z^2+x^2 + z^2-y^2 = 14
4(x^3+y^3) + z^3-x^3 = 195
To find the values of x, y, and z in the system of equations:
Step 1: Start with the first equation: 2x + y - z = 8.
Step 2: Continue with the second equation: x^2 - y^2 + 2z^2 = 14.
Rearrange this equation to solve for x^2:
x^2 = y^2 - 2z^2 + 14.
Step 3: Substitute the value of x^2 from Step 2 into the third equation: 3x^3 + 4y^3 + z^3 = 195.
Substitute (y^2 - 2z^2 + 14) for x^2:
3(y^2 - 2z^2 + 14)x + 4y^3 + z^3 = 195.
Step 4: Simplify the equation from Step 3:
3y^2x - 6z^2x + 42x + 4y^3 + z^3 - 195 = 0.
Step 5: Now, the system of equations is:
2x + y - z = 8,
3y^2x - 6z^2x + 42x + 4y^3 + z^3 - 195 = 0.
To solve this system of equations, further information is needed.
To find the values of x, y, and z in the given system of equations:
1. Solve the first equation: 2x + y - z = 8, for one variable.
- Express z in terms of x and y: z = 2x + y - 8.
2. Substitute the expression for z from step 1 into the second equation: x^2 - y^2 + 2z^2 = 14.
- Replace z with 2x + y - 8: x^2 - y^2 + 2(2x + y - 8)^2 = 14.
3. Simplify the equation obtained in step 2 and express it in terms of one variable.
- Expand and combine like terms: x^2 - y^2 + 2(4x^2 + y^2 + 64 + 4xy - 16x - 16y) = 14.
- Simplify further: x^2 - y^2 + 8x^2 + 2y^2 + 128 + 8xy - 32x - 32y = 14.
- Combine like terms: 9x^2 + y^2 + 8xy - 32x - 32y + 128 = 14.
- Rearrange terms: 9x^2 + y^2 + 8xy - 32x - 32y + 114 = 0.
4. Solve the equation obtained in step 3 for one variable, let's solve it for y in terms of x:
- Rearrange terms: y^2 + 8xy - 32y = -9x^2 + 32x - 114.
- Complete the square by adding (8x)^2 = 64x^2 to both sides:
y^2 + 8xy + 64x^2 - 32y + 64x^2 = -9x^2 + 32x - 114 + 64x^2.
- Factor the left side: (y + 8x)^2 - 32(y + 8x) = -9x^2 + 32x - 114 + 64x^2.
- Simplify further: (y + 8x)^2 - 32(y + 8x) = 55x^2 + 32x - 114.
- Rewrite as a binomial equation: (y + 8x)^2 - 32(y + 8x) - (55x^2 + 32x - 114) = 0.
5. Solve the quadratic equation obtained in step 4 for the variable (y + 8x). This can be done using factoring, the quadratic formula, or completing the square.
6. Once you have the expression for (y + 8x), substitute it back into the first equation to find z:
- Recall that z = 2x + y - 8.
Now you have expressions for y and z in terms of x, allowing you to find the solution to the system equation by substituting these values back into any of the original equations.