Let's analyze each description and match it with the appropriate linear equation.

1. For the line that goes through points $(-3, 1)$ and $(7, 6)$:

   • First, calculate the slope $m$ using the formula: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - 1}{7 - (-3)} = \frac{5}{10} = \frac{1}{2}$
   • Now, using the point-slope form $y = mx + b$ and substituting $m = \frac{1}{2}$ and one of the points, say $(-3, 1)$: $1 = \frac{1}{2}(-3) + b \Rightarrow 1 = -\frac{3}{2} + b \Rightarrow b = 1 + \frac{3}{2} = \frac{5}{2}$
   • Thus, the equation of the line is: $y = \frac{1}{2}x + \frac{5}{2}$

2. For the table of points $(0, 4)$ and $(5, 0)$:

   • Calculate the slope $m$ using the points $(0, 4)$ and $(5, 0)$: $m = \frac{0 - 4}{5 - 0} = -\frac{4}{5}$
   • Now, using the point-slope form with the point $(0, 4)$: $y = -\frac{4}{5}x + 4$
   • Thus, the equation of the line is: $y = -\frac{4}{5}x + 4$

3. For the line with a slope of $-\frac{1}{2}$ and a $y$-intercept of $-7$:

   • The slope-intercept form is straightforward: $y = -\frac{1}{2}x - 7$

Matching Answers

Based on these calculations, here’s the matching for each description:

1. A line goes through points $(-3, 1)$ and $(7, 6)$: $y = \frac{1}{2}x + \frac{5}{2}$
2. The table with points $(0, 4)$ and $(5, 0)$: $y = -\frac{4}{5}x + 4$
3. A line has a slope of $-\frac{1}{2}$ and a $y$-intercept of $-7$: $y = -\frac{1}{2}x - 7$

Would you like further details on these calculations or have any questions?

Here are some related questions to expand your understanding:

1. How can you determine the slope if you are given only the graph of a line?
2. What is the significance of the $y$-intercept inthe context of linear functions and graphs?
3. How would the equation change if the line had the same slope but passed through a different point?
4. How can you determine the equation of a line if only given its slope and one point on the line?
5. What are alternative methods to verify the accuracy of a line equation derived from two points?

Tip: When given two points, using the point-slope form of a line equation is a reliable method to derive the equation accurately, especially when working with fractional slopes.