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The image you have shared contains several mathematical problems related to linear systems and matrices. Here's a breakdown of the problems shown:

### Problem 6:
- **Part (a):** Determine the linear equation in variables \(x\) and \(y\) that has the general solution \(x = 5 + 2t\), \(y = t\).
- **Part (b):** Show that \(x = t\), \(y = \frac{1}{2}(t - 5)\) is also a general solution of the same system from part (a).

### Problem 7:
- The curve \(y = ax^2 + bx + c\) passes through the points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\). Show that the coefficients \(a\), \(b\), and \(c\) can be determined by solving a linear system based on the enlarged matrix representation:
  \[
  \begin{bmatrix}
  x_1^2 & x_1 & 1 & y_1 \\
  x_2^2 & x_2 & 1 & y_2 \\
  x_3^2 & x_3 & 1 & y_3
  \end{bmatrix}
  \]

### Problem 8:
- Consider the system of equations:
  \[
  \begin{aligned}
  x + y + 2z &= a \\
  x + z &= b \\
  2x + y + 3z &= c
  \end{aligned}
  \]
  Show that for this system to be consistent, the constants \(a\), \(b\), and \(c\) must satisfy \(c = a + b\).

### Problem 9:
- Show that if the linear equations \(x_1 + kx_2 = c\) and \(x_1 + lx_2 = d\) have the same solution set, then the two equations are identical.

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Which problem would you like to work through first, or would you like a detailed solution to one of these? Feel free to ask for clarification or further details.

Here are 5 related questions you might consider:
1. How do you determine the general solution to a linear system of equations?
2. What methods are commonly used to solve a system of quadratic equations through matrices?
3. How can consistency in a system of linear equations be verified?
4. What does it mean for two equations to have the same solution set?
5. How can you derive the conditions that make two linear equations identical?

**Tip:** Always verify the determinant when solving systems involving matrices to ensure there is a unique solution.

The image shows several mathematical problems related to linear systems and matrices. The key problems are: (a) Determining the linear equation for x and y with a general solution x = 5 + 2t, y = t, and showing that x = t, y = (1/2)(t - 5) is another solution. (b) Solving for coefficients of a quadratic equation y = ax^2 + bx + c passing through three points using a matrix. (c) Showing the consistency of a system of equations x + y + 2z = a, x + z = b, and 2x + y + 3z = c, requiring c = a + b. (d) Proving that two linear equations with the same solution set are identical.

This page provides a step-by-step guide to solving linear systems, quadratic equations using matrix methods, and verifying system consistency. The key concepts include determining solutions to linear equations and proving consistency in systems of equations using matrices.

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