Paige claims that the solution to the system of linear equations y = 2x - 1 and - 12x + 3y = 9 is (-2, -5). Select all methods that could be used to prove Paige's claim. A. Solve 3 + 4x = 2x - 1 for æ and substitute into y = 2x - 1 to find the value of y. B. Graph each linear equation and show that the intersection of the lines is (-2, -5). C. Solve - 12x + 3(2x - 1) = 9 for 2 and substitute into y = 2x - 1 to find the value of y. D. Substitute -5 for 2 and 2 for y into both equations and show that the equations are true.

To determine which methods could be used to verify Paige's claim that \((-2, -5)\) is the solution to the system of equations, we can analyze each option:

Given equations:
1. \( y = 2x - 1 \)
2. \( -12x + 3y = 9 \)

We need to check the methods provided.

**Option A**: "Solve \(3 + 4x = 2x - 1\) for æ and substitute into \(y = 2x - 1\) to find the value of y."
- This option is unrelated to the given system of equations. The equation \(3 + 4x = 2x - 1\) does not appear to be connected to the system, so it does **not** apply.
- **Incorrect**.

**Option B**: "Graph each linear equation and show that the intersection of the lines is \((-2, -5)\)."
- Graphing the lines \(y = 2x - 1\) and \( -12x + 3y = 9 \) is a valid way to check if their intersection point is \((-2, -5)\).
- **Correct**.

**Option C**: "Solve \( -12x + 3(2x - 1) = 9 \) for \(x\) and substitute into \(y = 2x - 1\) to find the value of \(y\)."
- This method uses substitution, replacing \(y\) in the second equation with \(2x - 1\) from the first equation and solving for \(x\), then finding \(y\). This is a valid method.
- **Correct**.

**Option D**: "Substitute \(-5\) for \(y\) and \(-2\) for \(x\) into both equations and show that the equations are true."
- This method tests if \((-2, -5)\) satisfies both equations by direct substitution, which is a valid approach.
- **Correct**.

### Final Answer:
- **Correct methods**: **B**, **C**, and **D**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you graph a system of linear equations to find their intersection point?
2. What is the substitution method for solving a system of linear equations?
3. How can you verify a solution by substitution?
4. Can all systems of linear equations be solved graphically?
5. What happens when two lines in a system are parallel?

**Tip**: Substituting the coordinates into both equations is often the quickest way to verify if a point is a solution to the system.

Paige claims that the solution to the system of linear equations y = 2x - 1 and -12x + 3y = 9 is (-2, -5). Select all methods that could be used to prove Paige's claim. A. Solve 3 + 4x = 2x - 1 for æ and substitute into y = 2x - 1 to find the value of y. B. Graph each linear equation and show that the intersection of the lines is (-2, -5). C. Solve -12x + 3(2x - 1) = 9 for 2 and substitute into y = 2x - 1 to find the value of y. D. Substitute -5 for 2 and 2 for y into both equations and show that the equations are true.

Discover how to verify Paige's claim that (-2, -5) is the solution to the system of equations y = 2x - 1 and -12x + 3y = 9. Learn graphing methods, substitution techniques, and how to check solutions for linear equations. Suitable for students in Grades 8-10.

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