Write an equation of the line that passes through the given point and is parallel to the graph of the given equation. Problem 1: (-4,-1); y = 2x + 14Problem 2: (6,2); y = (2/3)x + 19Problem 3: (-3,4); y = 2

Let's solve each of these one by one.

Problem 1: (-4,-1); y = 2x + 14 To write an equation of a line that is parallel to a given line and passes through a specific point, we need to know two things: the slope of the line and the given point through which the new line will pass.

The slope of the given line y = 2x + 14 is 2 (since the coefficient of x is the slope in the slope-intercept form y = mx + b). A parallel line will have the same slope. So, the slope of the line we wish to find is also 2.

Now we’ll use the point-slope form of a line which is y - y1 = m(x - x1), where m is the slope and (x1, y1) is the given point. Substituting the point (-4, -1) and the slope 2, we get:

y - (-1) = 2(x - (-4))

Simplify it to get the equation of the line:

y + 1 = 2(x + 4)

Now, distribute the 2:

y + 1 = 2x + 8

Subtract 1 from both sides to write it in slope-intercept form:

y = 2x + 7

So, the equation of the line parallel to y = 2x + 14 and passing through the point (-4, -1) is y = 2x + 7.

Problem 2: (6,2); y = (2/3)x + 19 Using the slope of y = (2/3)x + 19 which is 2/3, and the point (6, 2), we apply the point-slope form:

y - 2 = (2/3)(x - 6)

Multiplying 2/3 through, we get:

y - 2 = (2/3)x - (2/3)*6 y - 2 = (2/3)x - 4

Adding 2 to both sides, we have:

y = (2/3)x - 2

Therefore, the equation of the parallel line passing through (6, 2) is y = (2/3)x - 2.

Problem 3: (-3,4); y = 2 The equation y = 2 represents a horizontal line where the slope is 0, because it does not depend on x (slope m = 0 in the form y = mx + b). So, the line parallel to y = 2 must also be horizontal and therefore have a slope of 0.

The given point is (-3, 4). Using the point-slope form with a slope of 0 gives us:

y - 4 = 0(x - (-3))

This simplifies to:

y - 4 = 0

Adding 4 to both sides:

y = 4

The equation of the line parallel to y = 2 and passing through (-3,4) is y = 4.