I’m not sure how to do these Geometry questions

Question

Write the equation of the line parallel to y=2x+10 through the point (6,4)

Write the equation of the line perpendicular to y=2x+10 through the point (1,4)

Answer 1

Here are some crucial things for these questions

1. If two lines are parallel, they have the same slop.
2. If two lines are perpendicular of each other, the product of their slops equal to -1 (e.g. s1 * s2 = -1)

To solve first question, since we're trying to find the line that is parallel to y = 2x + 10, and y = 2x + 10 has slop of 2. So the line we're looking for also has slop of 2, therefore it can be represented by y = 2x + b, and b is a constant that we will need to find.

Another helpful information is it goes through point (6, 4), so we can simply plug x = 6, y = 4 to y = 2x + b. So we have 4 = 2 * 6 + b. Subtract 2*6 from both sides of the equations, we have 4 - 2*6 = b, so b = 4 - 12 = -8.

Now we know b, therefore, the line we're looking for is y = 2x - 8

Regarding to the second question,

Because if two lines are perpendicular of each other, the product of their slops equal to -1 (e.g. s1 * s2 = -1). Assume the slop of the line we're looking for is s. so we have s * 2 = -1, so s = -1/2. Because we know the slop, the line can be represented as y = (-1/2) x + b. The line also goes through point (1, 4), so similar to first question, we can plug in x = 1, y = 4 to y = (-1/2) x + b to find b. Then we have:

4 = (-1/2) * 1 + b

=> 4 = -1/2 + b, add 1/2 to both sides

=> 4 + 1/2 = b

=> b = 9/2

so the equation of the line we're looking for is y = (-1/2) x + 9/2.

Feel free to reach out to me if you have further question.

Answer 2

y = 2x + 10 @ point (6,4)

To find the parallel line from the original equation, we must know the slope and the point (already given). The slope is 2. Then you will need the point-slope form to find the new equation.

y - 4 = 2(x - 6)

y - 4 = 2x - 12

y = 2x - 8

y = 2x + 10 @ point (1,4)

To find the perpendicular line from the original equation, we must know the negative reciprocal of the original slope. In this case, we have -1/2. The point is already given to us. We need the point-slope form to find the new equation.

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

y - 4 = (-1/2)x + 1/2 (Add 4 and 1/2 will give us 4 and 1/2, or 9/2)

y = (-1/2)x + 9/2

Answer 3

y=2x+10.

The given equation is in the form y=mx+c.

So, m is the slope : m=2

If a line, let's say L1 is parallel to the y=2x+10. Then, the slope of L1 is same as the slope of the line y=2x+10.

Note this:

1) When two lines are parallel, their slopes are equal. : m1 = m2

2) When two lines are perpendicular, the product of the slopes is -1 : m1 * m2 = -1

So, by applying the above points , we get,

y=2x+10

m=2, (6 , 4 )

(x1,y1)

When we have slope and point, the equation is : y - y1 = m(x - x1)

y - 4 = 2 (x - 6)

y - 4 = 2x - 12

y = 2x - 8

perpendicular : m1 = 2 for euation : y=2x+10

m1 * m2 = -1

2 * m2 = -1

m2 = -1/2 , (1 , 4)

(x1,y1)

When we have slope and point, the equation is : y - y1 = m(x - x1)

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

y = (-1/2)x + 9/2

If you further have any doubts,you can message me, i am available anytime to clarify.