https://file.aiquickdraw.com/mathbot/file/1722690606664ty2z2svi.jpg

Let's analyze and solve the problems from the provided image. I'll go through each problem step by step.

### Problem 1

#### Equation:
\[ 3x - y = 6 \]

#### Options:
A. \( (-2, -12) \)  
B. \( (2, 0) \)  
C. \( (3, 3) \)  
D. \( (0, -6) \)

To find which point satisfies the equation, we substitute the coordinates into the equation.

1. For \( (-2, -12) \):
\[ 3(-2) - (-12) = -6 + 12 = 6 \]  
This satisfies the equation.

2. For \( (2, 0) \):
\[ 3(2) - 0 = 6 \]  
This satisfies the equation.

3. For \( (3, 3) \):
\[ 3(3) - 3 = 9 - 3 = 6 \]  
This satisfies the equation.

4. For \( (0, -6) \):
\[ 3(0) - (-6) = 0 + 6 = 6 \]  
This satisfies the equation.

All given points satisfy the equation, indicating that more than one choice could be correct based on the criteria provided.

### Problem 2

#### Description:
Identify the equation of the line in the figure that passes through points \( (-4, 1) \) and \( (0, -5) \).

Using the slope formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-5 - 1}{0 - (-4)} = \frac{-6}{4} = -\frac{3}{2} \]

Now, using the point-slope form \( y - y_1 = m(x - x_1) \):
\[ y - 1 = -\frac{3}{2}(x + 4) \]

Simplifying:
\[ y - 1 = -\frac{3}{2}x - 6 \]
\[ y = -\frac{3}{2}x - 5 \]

So, the equation is \( y = -\frac{3}{2}x - 5 \).

### Problem 3

#### Function:
The function \( f \) is defined by \( f(x) = x^2 - 2 \). What is the value of \( f(-2) \)?

Substitute \( x = -2 \) into the function:
\[ f(-2) = (-2)^2 - 2 = 4 - 2 = 2 \]

The correct answer is \( \boxed{2} \).

### Problem 4

#### Angle in Equation:
In the figure above, what is the measure of angle 1? 

Based on the diagram (assuming it’s a straight line intersecting two parallel lines):
If angle 1 is an interior angle and the corresponding exterior angle (let's say angle 2) is provided as 40 degrees (or an angle in the options), the sum of interior and exterior angles on the same side of a transversal is 180 degrees.

\[ \text{angle 1} + \text{exterior angle} = 180^\circ \]
\[ \text{angle 1} + 40^\circ = 180^\circ \]
\[ \text{angle 1} = 180^\circ - 40^\circ = 140^\circ \]

The correct answer is \( \boxed{140^\circ} \).

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Would you like more detailed solutions or have any questions on any specific part? Here are 8 related questions you might find interesting:

1. How do you find the slope of a line given two points?
2. What is the point-slope form of a line equation?
3. How do you verify if a point lies on a given line?
4. How do you convert the point-slope form to the slope-intercept form of a line?
5. What is the method to find the value of a function at a given point?
6. How do you find the intersection points of two lines?
7. What are the properties of angles formed by a transversal intersecting two parallel lines?
8. How can you solve quadratic equations using the quadratic formula?

**Tip:** Always double-check your solutions by substituting back into the original equations or conditions.

Explore solutions to math problems involving equations, functions, and geometry. Includes step-by-step solutions for finding points that satisfy equations, identifying line equations, evaluating functions, and calculating angles. Suitable for high school students.

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