To solve these problems, we need to find the equations of lines that are parallel to the given lines and pass through the specified points. Let's handle each problem step by step.

1. Equation of the line parallel to $y = 3x + 2$ and passing through $(1, 2)$:

A line parallel to another line has the same slope. The slope of the given line $y = 3x + 2$ is 3. Therefore, the equation of the parallel line will also have a slope of 3, and it will take the form:

$y - y_1 = m(x - x_1)$

Where:

• $m = 3$ (the slope),
• $(x_1, y_1) = (1, 2)$ is the point the line passes through.

Substitute these values into the point-slope form:

$y - 2 = 3(x - 1)$

Simplify:

$y - 2 = 3x - 3$

$y = 3x - 1$

Thus, the equation of the line parallel to $y = 3x + 2$ and passing through $(1, 2)$ is:

$y = 3x - 1$

2. Equation of the line parallel to $x + 2y = 6$ and passing through $(4, 3)$:

First, rewrite the equation $x + 2y = 6$ in slope-intercept form (i.e., solve for $y$):

$2y = -x + 6$

$y = -\frac{1}{2}x + 3$

The slope of this line is $-\frac{1}{2}$, so the equation of the parallel line will also have a slope of $-\frac{1}{2}$. Using the point-slope form with $(x_1, y_1) = (4, 3)$:

$y - y_1 = m(x - x_1)$

Substitute the values:

$y - 3 = -\frac{1}{2}(x - 4)$

Simplify:

$y - 3 = -\frac{1}{2}x + 2$

$y = -\frac{1}{2}x + 5$

Thus, the equation of the line parallel to $x + 2y = 6$ and passing through $(4, 3)$ is:

$y = -\frac{1}{2}x + 5$

Would you like further details or have any questions on the steps?

Related questions:

1. What is the difference between parallel and perpendicular lines in terms of their slopes?
2. How can you convert a general linear equation to slope-intercept form?
3. What is the significance of the point-slope form in line equations?
4. Can a vertical line be written in point-slope form, and how would it change?
5. How do you find the distance between two parallel lines?

Tip:

Always remember that parallel lines have equal slopes, while perpendicular lines have slopes that are negative reciprocals of each other.