Question 1186000
Here's how to solve the system of equations using the methods you requested:

**1. Graphing Method:**

1.  **Rewrite the equations in slope-intercept form (y = mx + b):**
    *   126x + 198y = 150  =>  y = (-126/198)x + (150/198)  => y ≈ -0.636x + 0.758
    *   338x + 43y = 303  =>  y = (-338/43)x + (303/43)  => y ≈ -7.86x + 7.047

2.  **Plot the lines:** Graph both equations on the same coordinate plane.  The point where the lines intersect is the solution.

3.  **Approximate the solution:** From the graph, you'll see that the lines intersect somewhere around x ≈ 0.8 and y ≈ 0.3.  Graphing isn't always perfectly precise, so these are just estimates.

**2. Substitution Method:**

1.  **Solve one equation for one variable:**  Let's solve the first equation for y:
    y = (-126/198)x + (150/198)

2.  **Substitute:** Substitute this expression for y into the second equation:
    338x + 43((-126/198)x + (150/198)) = 303

3.  **Simplify and solve for x:**
    338x - (5418/198)x + (6450/198) = 303
    (66864/198)x - (5418/198)x = (60054/198) - (6450/198)
    (61446/198)x = (53604/198)
    x = 53604 / 61446
    x ≈ 0.872

4.  **Substitute x back to find y:** Substitute the value of x into either of the original equations. Using the first equation is simpler.
    126(0.872) + 198y = 150
    110 + 198y = 150
    198y = 40
    y ≈ 0.202

**3. Elimination Method:**

1.  **Multiply equations to match coefficients:** We want to eliminate either x or y.  Let's eliminate y. Multiply the first equation by 43 and the second equation by 198:
    *   (126x + 198y = 150) * 43  =>  5418x + 8514y = 6450
    *   (338x + 43y = 303) * 198  =>  66864x + 8514y = 60054

2.  **Subtract the equations:** Subtract the first new equation from the second:
    61446x = 53604

3.  **Solve for x:**
    x ≈ 0.872

4.  **Substitute x back to find y:** Substitute the value of x into either of the original equations.
    126(0.872) + 198y = 150
    y ≈ 0.202

**4. Determinant Method (Cramer's Rule):**

1.  **Set up the coefficient matrix (D) and the matrices for x (Dx) and y (Dy):**

    D = | 126  198 |
        | 338   43 |

    Dx = | 150  198 |
         | 303   43 |

    Dy = | 126  150 |
         | 338  303 |

2.  **Calculate the determinants:**
    *   det(D) = (126 * 43) - (198 * 338) = -61446
    *   det(Dx) = (150 * 43) - (198 * 303) = -53604
    *   det(Dy) = (126 * 303) - (150 * 338) = -12300

3.  **Solve for x and y:**
    x = det(Dx) / det(D) = -53604 / -61446 ≈ 0.872
    y = det(Dy) / det(D) = -12300 / -61446 ≈ 0.200

**Solution:**

The solution using all methods converges to approximately:

*   x ≈ 0.872
*   y ≈ 0.202