To match each rate of change (slope) and initial value (y-intercept) to a pair of points, we can use the equation of a line in the form:

$y = mx + b$

where $m$ is the rate of change (slope) and $b$ is the initial value (y-intercept). We can verify which pairs match each rate of change and initial value by substituting the points to see if they satisfy the line equation.

Steps

1. For each rate of change and initial value, we form an equation and check which point pairs satisfy the equation.

1. Rate of change = -1, initial value = 7

Equation: $y = -x + 7$

• Testing (4, 3) and (6, 7):

  • For (4, 3): $y = -4 + 7 = 3$
  • For (6, 7): $y = -6 + 7 = 1$
  • No match.

• Testing (2, 10) and (5, 19):

  • For (2, 10): $y = -2 + 7 = 5$
  • No match.

• Testing (2, 5) and (5, 2):

  • For (2, 5): $y = -2 + 7 = 5$
  • For (5, 2): $y = -5 + 7 = 2$
  • Match found.

2. Rate of change = 2, initial value = -5

Equation: $y = 2x - 5$

• Testing (4, 3) and (6, 7):
  • For (4, 3): $y = 2(4) - 5 = 3$
  • For (6, 7): $y = 2(6) - 5 = 7$
  • Match found.

3. Rate of change = -0.5, initial value = 5

Equation: $y = -0.5x + 5$

• Testing (2, 4) and (4, 3):
  • For (2, 4): $y = -0.5(2) + 5 = 4$
  • For (4, 3): $y = -0.5(4) + 5 = 3$
  • Match found.

4. Rate of change = 3, initial value = 4

Equation: $y = 3x + 4$

• Testing (2, 10) and (5, 19):
  • For (2, 10): $y = 3(2) + 4 = 10$
  • For (5, 19): $y = 3(5) + 4 = 19$
  • Match found.

Final Matches

• Rate of change = -1, initial value = 7: (2, 5) and (5, 2)
• Rate of change = 2, initial value = -5: (4, 3) and (6, 7)
• Rate of change = -0.5, initial value = 5: (2, 4) and (4, 3)
• Rate of change = 3, initial value = 4: (2, 10) and (5, 19)

Would you like further explanations or have any questions?

Additional Questions

1. How is the slope calculated between two points?
2. What are the steps to determine if a point lies on a line with a given equation?
3. How does changing the slope affect the steepness of a line?
4. What is the significance of the y-intercept in a linear equation?
5. How would you determine the equation of a line given two points?

Tip

When matching points to a line equation, substitute each point into the equation to confirm that both satisfy it.