Let's break down and solve the problems shown in the image step by step:

1. First Equation: $y = -5x + 2$

• Slope: The equation is already in slope-intercept form $y = mx + b$, where:

  • $m$ is the slope.
  • $b$ is the y-intercept.
  • Here, $m = -5$ and $b = 2$.

  Answer:

  • Slope $= -5$
  • Y-intercept $= 2$

2. Second Equation: $y = -x - 10$

• Slope: Again, this equation is in slope-intercept form $y = mx + b$.

  • The slope $m = -1$, because $y = -x$ is the same as $y = -1x$.
  • The y-intercept $b = -10$.

  Answer:

  • Slope $= -1$
  • Y-intercept $= -10$

3. Write an Equation Using the Points (0, 2) and (7, 7)

To write the equation of the line, we need the slope and y-intercept.

• Find the Slope: Using the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, where:

  • $(x_1, y_1) = (0, 2)$
  • $(x_2, y_2) = (7, 7)$

  $m = \frac{7 - 2}{7 - 0} = \frac{5}{7}$

• Find the Equation: The y-intercept is given by the point $(0, 2)$, so $b = 2$. Using the slope-intercept form $y = mx + b$:

  • $y = \frac{5}{7}x + 2$

  Answer: The equation is $y = \frac{5}{7}x + 2$.

Would you like more details or clarifications on any of the steps?

Here are some follow-up questions to consider:

1. What is the slope if the equation is written in standard form $Ax + By = C$?
2. How do you find the slope between two points if one of the points is negative?
3. What happens to the graph of a line when the slope is increased or decreased?
4. Can you explain how to convert from slope-intercept form to point-slope form?
5. How would the equation change if the y-intercept were negative?

Tip: Always remember, the slope represents the "steepness" of the line. A positive slope means the line rises, while a negative slope means the line falls.