Analytical Geometry Questions (3 questions, I'm really struggling with this unit.)

Question

FIRST QUESTION:

 

I've tried attempting this but I'm unsure of my answers. Can someone explain to me?

 

The question I am struggling with is asking me to determine which of the following lines is the steepest and explain why.

 

a. x+2y+8=0

b. y= 3x+5

c. 10=8x+2y

 

I need help with rewriting these lines into slope-intercept form. (is this right and I supposed to be doing this? or is there a different way to solve this?)

 

SECOND QUESTION:

 

Find the x and y intercepts of the following relations:

 

a. 3x - 5y = 45

b. -6x = y - 18

c. 4x + 6y - 17=0

 

I have already solved 'a' so far, x intercept is (15,0) and the y intercept is (0,9), is this correct?

 

THIRD QUESTION:

 

Find the equations of the following lines:

 

a.having slope of -2 and passing through the point (-4, 2)

 

b.parallel to y=6x+3 and passing through the point (-1, 13)

 

c.perpendicular to 3x-5y and having an x-intercept of 3

 

d.passing through (-2, -6) and (4, 2)

 

e.having an x-intercept of -3 and a y-intercept of 6

 

 

 

 

 

Answer 1

Hello Mickayla,

 

Interesting questions.  Let's explore them a bit.

 

FIRST question.  To determine steepness the slope of the line is essential.  So the slope-intercept form of the equation is the most natural place to start unless we are given two points.  To reach the simplest slope-intercept form we isolate y on the left side of the equality and simplify the right side reaching the form y = mx - c  where m is slope of the line.

 

a. x + 2y + 8 = 0  --->  2y = -x - 8  Note that x has an implied coefficient of -1 and between x and 8 is a subtraction sign.  It helps to think of the overloaded operator '-' as which of its applications, polarity or arithmetic, are in use.

2y = -x + 8 is divided on both sides by 2 to yield  y = (-1/2)x + 4  So m = -1/2 = slope

 

b. y = 3x + 5 Here m = 3 = slope

 

c. 10 = 8x + 2y  ---> -2y +10 = 8x  ---> -2y = 8x - 10 ---> y = -4x + 5  So m = -4 = slope

 

The slopes of the three given lines are -1/2, 3, and -4.  For this question only the magnitude of the slope is interesting, not the sign (or polarity), so equation c has the steepest slope.

 

SECOND question:

Find x and y intercepts for relations.  While all equalities are a subset of relations, these equations are equalities so we use only the rules for equalities.

 

a. 3x - 5y =45  --->  -5y = -3x + 45  --->  y = 3x/5 - 9  To determine intercepts we set the other variable to 0 and simplify.  For x =0, y = -9.  This is the point (0, -9) which is the y intercept.

Now we set y =0 and 0 = 3x/5 -9 ---> 9 =3x/5 ---> 45 = 3x ---> 15 = x and the point (15, 0) is the x intercept.

 

b. -6x = y - 18  Using the reflexive property of equalities y -18 = -6x  ---> y = -6x +18.  Now set x = 0 to find the y intercept and then y = 0 to find the x intercept.  (Do it.) You should get (0, 18) and (3, 0).

 

c. 4x + 6y -17 = 0 ---> 6y = -4x + 17  ---> y = -2x/3 + 17/6.

Doing the substitutions you should get (17/4, 0) and (0, 17/6).

 

THIRD question:

 

a.  Starting with the general slope intercept form y = mx +c ---> y = -2x +c  Now substitution will give the value for c

 

    2 = -2(-4) + c  --->2 = 8 + c ---> c = -6.  Therefore y = -2x -6  Checking with the point (-4, 2) shows this is the correct answer.

 

b.  Since it will be parallel to the given line, they must have the same slope: m = 6  Using the general form again to find c,   13 =6(-1) + c   --->  c = 19  AND  y = 6x +19.  NOW check the answer using the point (-1, 13).  Be sure you can fill in all the steps for this part that I have skipped.  If yo understand the details the rest will be much easier.

 

c.  Perpendicular.  Skipping the derivation of the concept, the slope of a perpendicular line will have the negative inverse of the slope of the line to which it is perpendicular.  If the original line has slope m then a line perpendicular to it will have a slope of -1/m.  So with a slope of  3/5 the perpendicular slope will be -5/3.  y = -5x/3 + c  Substituting the intercept (3, 0) yields c= 5 and the line is y = -5x/3 + 5  (Remember to supply each of the steps I am skipping here.  You will learn it much easier by doing all the required steps and be able to apply it next time.)

 

d.  To get the slope with two points given, we use the definition of slope:  m = (y2 - y1) / (x2 - x1)  Either point can be (x1, y1) as long as you use them consistently.  So write down (x1, y1) = "your choice" on your paper to avoid confusion as you work though each problem.  Here m = (2 - (-6)) / (4 - (-2)) = 8/6 = 4/3

y = 4x/3 + c  Working now with the first point:  -6 = 4(-2)/3 + c and c = -6 + 8/3 = (-18 +8)/3 = -10/3

y = 4x/3 -10/3 .  Now check with the original points for accuracy (be certain to do this yourself).

 

e.  Two points are given in intercept form.  They are (-3, 0) and (0, 6)  From here this is solved in exactly the same way as part d above.  Your result should be:  m = 2 and y = 2x + 6.  Check that this answer is correct using the given points.

 

Please feel free to contact me if you need additional detail.

 

Best wishes Mickayla.  JAG

Answer 2

Let's take these one at a time.

 

First question:  To find slope form an equation, make sure that it is in slope intercept form by solving for y.  The value in front of the x is automatically the slope.

 

a.

x + 2y + 8 = 0

2y = -x - 8

y = (-1/2)x - 4

Slope: -1/2

 

b. 

y = 3x +5

Already in slope intercept form.  Slope = 3

 

c.

10 = 8x + 2y

2y = 8x - 10

y = 4x - 5

Slope = 4

 

4 is the largest of the slopes, so line c is the steepest.

 

Second question:  To find the x-intercept, let y = 0 and solve for x.  To find the y-intercept, let x = 0 and solve for y.

 

a.  3x - 5y = 45

x-int:

3x - 5(0) = 45

3x=45

x = 15

 

y - int:

3(0) - 5y = 45

-5y = 45

y = -9

 

Therefore the x-int is (15, 0) and y-int is (0, -9)  (You were close.)

 

b. -6x = y - 18

x-int:

-6x = 0 - 18

-6x = -18

x = 3

 

y-int:

-6(0) = y - 18

0 = y - 18

18 = y

 

Therefore the x-int is (3, 0) and the y-int is (0, 18)

 

c. 4x + 6y - 17 = 0

x-int:

4x + 6(0) - 17 = 0

4x - 17 = 0

4x = 17

x = 17/4 or 4.25

 

y-int:

4(0) + 6y - 17 = 0

6y - 17 = 0

6y = 17

y = 17/6 or approx. 2.83

 

Therefore the x-int is (17/4, 0) and the y-int is (0, 17/6)

 

Third question:  This is a little complex question because it goes is a number of directions.  However, the simplest way to write a linear equation in any situation is to use point-slope form (y - y₁ = m(x - x₁)) then simplify to slope-intercept form.

 

a. having a slope of -2 and passing through the point (-4, 2)

 

You have both slope and a point, so just plug in.

 

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

y - 2 = -2(x + 4)

y - 2 = -2x - 8

y = -2x - 6

 

b.parallel to y=6x+3 and passing through the point (-1, 13)

 

parallel lines have the same slopes, so your new line will also have a slope of 6.

 

y - 13 = 6(x - (-1))

y - 13 = 6(x + 1)

y - 13 = 6x + 6

y = 6x + 19

c.perpendicular to 3x-5y and having an x-intercept of 3

 

Perpendicular slopes are negative reciprocals of each other.  Find the slope of the first line, them flip the sign and flip the fraction.  Even though you didn't give me the full equation, I can still find the slope by solving 3x - 5y = C.

 

3x - 5y = C

-5y = -3x + C
y = (3/5)x - (1/5)C

 

The original slope is 3/5 so the perpendicular slope will be -5/3.  If the x-intercept is 3, the the point (3,0) is on the line.

 

y - 0 = -5/3(x - 3)

y = -5/3x + 5

d.passing through (-2, -6) and (4, 2)

 

First you calculate the slope.

 

(2 - (-6))/(4 - (-2)) = (2 + 6)/(4 +2) = 8/6 = 4/3

 

Pick either point to find your equation.  It doesn't matter which since they both will give the same result.

 

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

y + 6 = 4/3 (x + 2)

y + 6 = 4/3x + 8/3

y = 4/3x + 26/3

 

e.having an x-intercept of -3 and a y-intercept of 6

 

The x-int of -3 means (-3, 0) is a point on the line.  The y-int of 6 means (0, 6) is on the line.  We can use these to find the slope.

 

(6 - 0)/ (0 - (-3)) = 6/3 = 2

 

y - 0 = 2(x - (-3))

y = 2(x + 3)

y = 2x + 6

Answer 3

Question 1)

 

You need to get all of the choices in slope-intercept form (y=mx+b).  Then compare their slopes.  The largest slope contains the steepest line.

 

Question 2)

 

To find the x-intercept, set y=0 and solve for x.  x-intercepts are in the form (x, 0).

To find the y-intercept, set x=0 and solve for y.  y-intercepts are in the form (0, y).

 

Question 3)

 

Use point-slope form of the line for all parts in this question.

 

y = m(x - x₁) + y₁

 

where:

m = slope

(x₁ , y₂) is the point the line passes through

 

Note that parallel lines have the same slopes.  Perpendicular lines have negative reciprocal slopes.  Use these hints to answer all parts.