8x - 4y + 2z=16 for elimanation method,substitution method and cramer's rule?
where are the other equations?
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To solve the given equation 8x - 4y + 2z = 16 using different methods, let's understand each method step by step.
1. Elimination Method:
The elimination method involves eliminating one variable at a time by adding or subtracting equations. Here are the steps:
Step 1: Rearrange the equation to standard form:
8x - 4y + 2z = 16
Step 2: Choose two pairs of equations and eliminate a variable:
Let's assume another equation, for example, 2x + 3y + 4z = 10.
Multiply the first equation by a suitable factor to make the coefficients of variables "x" equal in both equations. Let's multiply the first equation by 2:
(2)(8x - 4y + 2z) = (2)(16)
16x - 8y + 4z = 32
Step 3: Subtract the modified first equation from the second equation:
(2x + 3y + 4z) - (16x - 8y + 4z) = 10 - 32
-14x + 11y = -22
Step 4: Solve the resulting equation:
Now, you have a new equation -14x + 11y = -22, which can be solved to find the values of "x" and "y".
2. Substitution Method:
In the substitution method, we solve one equation for one variable and substitute that value into another equation. Here's how it's done:
Step 1: Rearrange the equation to standard form:
8x - 4y + 2z = 16
Step 2: Choose one equation and solve it for one variable:
Let's assume another equation, for example, 2x + 3y + 4z = 10. Solve it for "x":
2x = 10 - 3y - 4z
x = (10 - 3y - 4z) / 2
Step 3: Substitute the expression for "x" into the first equation:
8((10 - 3y - 4z) / 2) - 4y + 2z = 16
Step 4: Solve the resulting equation:
Now, you have an equation with variables "y" and "z" only. Solve it to find the values of "y" and "z".
3. Cramer's Rule:
Cramer's Rule involves using determinants to solve a system of linear equations. Let's see how it works:
Step 1: Rearrange the equation to matrix form:
8x - 4y + 2z = 16 can be represented as:
| 8 -4 2 | | x | | 16 |
| | * | | = | |
| ... ... ... | | y | | ... |
Step 2: Find the determinant of the coefficient matrix (D)
| 8 -4 2 |
| | = D
| ... ... ... |
Step 3: Replace the column containing the variables with the constant terms and find the determinant (Dx) to solve for "x":
| 16 -4 2 |
| | = Dx
| ... ... ... |
Step 4: Similarly, find the determinants Dy and Dz by replacing the respective columns and solve for "y" and "z".
Step 5: Finally, divide the values of variables by the determinant of the coefficient matrix:
x = Dx / D,
y = Dy / D,
z = Dz / D.
These are the three methods you can use to solve the given equation 8x - 4y + 2z = 16. Choose the method that suits you best based on the given context.