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selesaikan spl ini dengan eliminasi metode gauss jordan

The problem asks to find the coefficients \( a \), \( b \), \( c \), and \( d \) for the equation of a circle \( ax^2 + ay^2 + bx + cy + d = 0 \), given the points \( (-2, 7) \), \( (-4, 5) \), and \( (4, -3) \), which lie on the circle.

### Step 1: Equation of the Circle
The general form of a circle is:

\[
ax^2 + ay^2 + bx + cy + d = 0
\]

Since the equation is for a circle, the coefficients of \( x^2 \) and \( y^2 \) must be equal, i.e., \( a = a \). So the equation simplifies to:

\[
x^2 + y^2 + bx + cy + d = 0
\]

Now, let's substitute the three points given on the circle into this equation.

### Step 2: Substituting Points

1. **For \( (-2, 7) \):**

\[
(-2)^2 + (7)^2 + b(-2) + c(7) + d = 0
\]
\[
4 + 49 - 2b + 7c + d = 0
\]
\[
53 - 2b + 7c + d = 0 \quad \text{(Equation 1)}
\]

2. **For \( (-4, 5) \):**

\[
(-4)^2 + (5)^2 + b(-4) + c(5) + d = 0
\]
\[
16 + 25 - 4b + 5c + d = 0
\]
\[
41 - 4b + 5c + d = 0 \quad \text{(Equation 2)}
\]

3. **For \( (4, -3) \):**

\[
(4)^2 + (-3)^2 + b(4) + c(-3) + d = 0
\]
\[
16 + 9 + 4b - 3c + d = 0
\]
\[
25 + 4b - 3c + d = 0 \quad \text{(Equation 3)}
\]

### Step 3: System of Equations
We now have a system of 3 equations:

1. \( 53 - 2b + 7c + d = 0 \)
2. \( 41 - 4b + 5c + d = 0 \)
3. \( 25 + 4b - 3c + d = 0 \)

### Step 4: Solve Using Gaussian Elimination (Gauss-Jordan Method)
I will now solve this system of equations using the Gaussian elimination method.

Let me calculate this.The solution to the system of equations is:

\[
b = -2, \quad c = -4, \quad d = -29
\]

Thus, the equation of the circle is:

\[
x^2 + y^2 - 2x - 4y - 29 = 0
\]

Would you like further details or clarification on any steps? 

Here are some related questions:
1. How would you apply the same method to solve for a different shape, like an ellipse?
2. What changes if we had additional points on the circle?
3. How does Gaussian elimination work for larger systems of equations?
4. What if the points were collinear—how would that affect the result?
5. How can we verify that the points truly lie on this circle?

**Tip:** Gaussian elimination is a powerful method for solving linear systems by transforming the matrix into a reduced row-echelon form.

Carilah koefisien a, b, c, dan d yang memenuhi persamaan lingkaran ax^2 + ay^2 + bx + cy + d = 0

This problem involves finding the coefficients a, b, c, and d for a circle equation, given points that lie on the circle. Using algebraic substitution and Gaussian elimination, solve for the unknowns. Suitable for students in Grades 10-12.

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