Write an equation in slope-intercept form for the line that passes through the given point and is parallel to the given line.

Question

1. (-3,5); y= -1/2 x + 4
2. (-7,3); x=4

Please explain what you did so I can learn! Thanks!

Answer 1

For problem 1, in the equation y = -1/2 x + 4, is given in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. So, the slope is given as -1/2, so this is the slope we need in our new equation. At this point we do not know what is the value of the y-intercept, or b, so we need to find that.

To find the y-intercept, or b, we plug in the values of the point that we want our line to go through (-3, 5) into

y = mx + b with -1/2 in as the slope, so we get

5 = -1/2 (-3) + b and we solve for b.

5 = 3/2 + b (we multiplied -1/2 times -3 to get 3/2)

10/2 = 3/2 + b (here we changed 5 to 10/2 so that we can next subtract 3/2)

10/2 - 3/2 = 3/2 - 3/2 + b (subtract 3/2 from both sides)

7/2 = b We have now solved for b

Now we take the equation y = mx + b and substitute -1/2 for the slope and 7/2 for b to get our new equation which is y = -1/2 x + 7/2

For Problem 2, we have the equation x = 4, which is a vertical line that passes through all points where

x = 4. A vertical line has an undefined slope, so the equation cannot be written in slope-intercept form. Our new equation will also have an undefined slope and will also be in the form x = ?. If the equation passes through the point (-7, 3), then it passes through all points where x = -7, which is our new equation.

Please ask if you do not understand and I will try to clarify.

Answer 2

'Parallel to the given line' means both lines must have the same slope.

We are looking for a line that runs through (-3.5) and has slope -1/2.

An aside, to help explain slope-intercept form--

We want slope-intercept form (y = mx + b). Slope-intercept form gives us a slope m and a value b. This value b represents the point along the y-axis through which the line passes. We can think of this as the coordinate along the line where x=0, as all y values where x=0 are on the y-axis. When x = 0, y = 0x + b, leading to y =b, leading to a coordinate along the line (0,b).

Back to the question. We know (-3,5) is on our line, and we want to find b such that (0,b) is on the line. We know that for every time x increases by 1, y decreases by 0.5. so if (-3, 5) is on line, so is (-2, 4.5), (-1, 4), and (0, 3.5). 3.5 is our b, and so we are done. y = -1/2x + 3.5.

A more formulaic y to approach this is using point-slope form. This gives an equation for a line using a point on the line and the slope, hence point-slope. Point slope is defined as y- y1 = m(x - x1), where (x₁, y₁) is our point and m is our slope. This becomes y - 5 = -1/2(x - (-3)). This simplifies: y - 5 = -1/2x -1.5. y = -1/2x +3.5. We can see this is the same as above.

In essence, using the point-slope formula is doing exactly what I did to find the equation the first time. It's important to understand why this works to build a good fundamental foundation.

#2 requires a more analytical approach. x=4 is the vertical line with x coordinate 4. Think of it as the sum of all points (4, y) where y ranges over all values.

We want the vertical line (to be parallel) through (-7,3). A vertical line is defined by an x-coordinate, and an x-coordinate only. Can you come up with the equation of this vertical line?