It looks like you are describing two lines, $l_1$ and $l_2$, each defined by two points:

• $l_1: (4, 6)$ and $(5, 7)$
• $l_2: (-1, -1)$ and $(1, 4)$

Let’s find the equations of these lines.

Step 1: Find the slope of each line

The slope $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

For line $l_1$:

The points are $(4, 6)$ and $(5, 7)$. $m_1 = \frac{7 - 6}{5 - 4} = \frac{1}{1} = 1$

For line $l_2$:

The points are $(-1, -1)$ and $(1, 4)$. $m_2 = \frac{4 - (-1)}{1 - (-1)} = \frac{5}{2}$

Step 2: Find the equation of the lines

We can use the point-slope form of a line equation: $y - y_1 = m(x - x_1)$

For line $l_1$:

Using point $(4, 6)$ and slope $m_1 = 1$: $y - 6 = 1(x - 4)$ Simplifying: $y - 6 = x - 4 \quad \Rightarrow \quad y = x + 2$

For line $l_2$:

Using point $(-1, -1)$ and slope $m_2 = \frac{5}{2}$: $y - (-1) = \frac{5}{2}(x - (-1))$ Simplifying: $y + 1 = \frac{5}{2}(x + 1)$ $y + 1 = \frac{5}{2}x + \frac{5}{2}$ $y = \frac{5}{2}x + \frac{3}{2}$

Final equations:

• The equation of line $l_1$: $y = x + 2$
• The equation of line $l_2$: $y = \frac{5}{2}x + \frac{3}{2}$

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Do you need further details or have any questions about these calculations? Here are some related questions:

1. How do you calculate the intersection point of two lines?
2. How can we determine if two lines are parallel or perpendicular?
3. What are the properties of the slope-intercept form of a linear equation?
4. Can you explain the concept of the point-slope form more deeply?
5. How would you convert the equation of a line into general form?

Tip: The slope $m$ gives you the direction of the line: a positive slope indicates an upward trend, while a negative slope indicates a downward trend.