Teacher just gave us answer... Not enough practice

Question

I have a test today and here are a few questions I have the answers to but I don't know how the teacher got there. He takes too long teaching one problem that at the end leaves us confused. Please help me understand the following:

 

1. Decide whether the pair of lines is parallel, perpendicular, or neither.

    3x-8y=-7 and 32x+12y=-7

 

2. Find the slope-intercept form of the line satisfying the conditions.

    m=7/4; through (0,3)

 

3. Find an equation of the line satisfying the conditions. Write the equation in slope-intercept form.

    Through (-3,8); perpendicular to -3x+4y=-23

 

4. Through (-6,7); parallel to 3x+7y=3

 

5. Find an equation of the line passing through the two points. Write the equation in standard form.

    (-7,-5) and (2,5)

 

Answer 1

1. Find their slope

Write line equations in standard slope intercept form y = mx + b, m is slope and b is y intercept.

-8y = -7 - 3x

8y = 3x + 7

y = 3/8x + 7/8  slope is 3/8

 

second line

12y = -32x - 7

y = -32/12x - 7/12

y = -8/3x - 7/12     slope is -8/3 which is negative inverse of first line so its perperdicular to first line.

 

2. Standard slope intercept form

 y = mx + b

 

y = 7/4 x + b

Since line passes through (0,3), substitute x=0, y= 3

 

3 = 7/4 . o    + b

b = 3

substitute value of b

 

y = 7/4x + 3

To simplyfy multiply entire equation by 4

4y = 7x + 3

4y - 7x = 3

 

3. Line is perpendicular to -3x + 4y = -23.  Find slope by re writing it in slope intercept form

 

4y = 3x - 23

y = 3/4 - 23/4  slope of given line is 3/4

 

Therefore slope of new line would be -4/3

 

Equation of new line y = -4/3x  + b

since it passes through (-3,8) substitute y = 8 and x = -3

8 = -4/3 . -3   + b

8 = 4 + b

b = 4

Equation of new line

y = -4/3x  + 4

To simplify multiply entire equation by 3

3y = -4x + 12

3y + 4x = 12

 

4. Find slope of given line 3x + 7y = 3

 

rewrite in slope intercept form

y = -3/7 x + 3/7 therefor slope of given line is -3/7

New line is parallel to given line so its slope is also = -3/7

Equation of new line

y = -3/7x + b

since it passes through (-6,7) substitute x = -6 and y = 7

7= -3/7. -6 + b

7 = 18/7 + b

b = 7 - 18/7 = 31/7

Therefor equatio of new line

y = -3/7x + 31/7

To simplify multiply entire equation by 7

7y = -3x +31

7y + 3x = 31

 

5. First find slope

 

m = (y2 - y1)/(x2 - X1)

m = (5-(-5))/(2-(-7)) = 10/9

Line equation

y = mx + b = 10/9x + b

since it passes through (choose any one point) (2,5) substitute x = 2, y = 5

5= 10/9. 2 + b

5= 10/9 +b

b= 5 - 20/9 = 25/9

Euation of new line

 

y = 10/9x + 25/9

 

To simplify multiply entire equation by 9

9y = 10x + 25

9y - 10x = 25

 

:)

 

 

Answer 2

Hi Grace;

I believe I had the same instructor.  In college, he would talk endlessly.  He once had his hands in his jacket pockets when he began waiving his hands around without realizing that he forgot his hands were still in his pockets.  It was quite a spectacle.

 

1. Decide whether the pair of lines is parallel, perpendicular, or neither.
3x-8y=-7 and 32x+12y=-7

Both equations are in standard form...

Ax+By=C, neither A nor B equal zero, and A is greater than zero.

Slope is -A/B...

-[(3)/(-8)]

A negative of a negative is positive...

3/8

Slope is -A/B...

-[(32)/(12)]

Both numbers are divisible by 4...

-8/3

One slope is the negative inverse of the other.

THESE ARE PERPENDICULAR LINES.

2. Find the slope-intercept form of the line satisfying the conditions.
m=7/4; through (0,3)

Slope-intercept equation is...

y=mx+b

m is the slope.

b is the y-intercept, the value of y when x=0.  This is the point provided.

y=(7/4)x+3

3. Find an equation of the line satisfying the conditions. Write the equation in slope-intercept form.
Through (-3,8); perpendicular to -3x+4y=-23

The equation is not in standard format because A is less than zero.  Let's fix that by multiplying both sides by -1...

(-1)(-3x+4y)=-23(-1)

3x-4y=23

It does not make a difference when establishing slope.  But I go by the rules because when you enter the higher levels of algebra, it will matter.

Slope is -A/B...

-[(3)/(-4)]

3/4

The slope of the line perpendicular to this is its negative inverse...

-4/3...

Slope-intercept form is...

y=mx+b

y=(-4/3)x+b

Let's plug-in the one point provided, (-3,8), to establish the y-intercept, b.

8=[(-4/3)(-3)]+b

Note how the 3 in the numerator and denominator cancels, -3/3=-1...

8=[(-4)(-1)]+b

A negative number multiplied by a negative number has a positive result...

8=4+b

Let's subtract 4 from both sides...

8-4=4-4+b

4=b

y=(-4/3)x+4

4. Through (-6,7); parallel to 3x+7y=3

The equation is in standard form.  The slope is -A/B.

-[(3)/(7)]=-3/7

The slope of the line parallel to this is the same.

y=(-3/7)x+b

Let's plug-in the one point provided...

7=[(-3/7)(-6)]+b

7=(18/7)+b

Let's convert 7 into 49/7...

49/7=(18/7)+b

Let's subtract 18/7 from both sides...

(49/7)-(18/7)=-(18/7)+(18/7)+b

31/7=b

y=(-3/7)x+(31/7)

5. Find an equation of the line passing through the two points. Write the equation in standard form.
(-7,-5) and (2,5)

Let's first establish slope.  Slope is change-of-y divided by change-of-x...

(y-y₁)/(x-x₁)

(-5-5)/(-7-2)

-10/-9

A negative number divided by a negative number has a positive result...

10/9

Standard formula is...

Ax+By=C, neither A nor B equal zero, and A is greater than zero...

Slope is -A/B.

Slope is -(10/9)

Because A must be greater than zero, this is...

10/-9

10x-9y=C

Let's plug-in one point provided to establish the value of C.  I randomly select the first, (-7,-5)...

[(10)(-7)]-[(9)(-5)]=C

-70+45=C

-25=C

10x-9y=-25

Let's use the other point provided, (2,5), to verify the result...

[(10)(2)]-[(9)(5)]=-25

20-45=-25

-25=-25

 

 

Answer 3

1.  convert both equations into y =m(slope)*x+b(y intercept) form

8y=3x+7.    12y=-32x-7

y=(3/8)x+7/8  Y=-(32/12)x-7/12

                       Y=(-8/3)x-7/12

 

Look at the slopes.  If lines are parallel the two slopes are the same; if the two slopes are the negative reciprocal of each other, ie, when multiplied together the result is -1, the lines are perpendicular.

The two lines here are perpendicular.

 

2. Y=mx+b

y=(7/4)x+b

substitute coordinates for point given, (0,3)

3=(7/4)*0+b ... B=3

y=(7/4)x+3

 

3.-  the slope of the lgiven line is 3/4.  See first problem for derivation of y=(3/4)x-23/4.

since you are looking for perpendicular line, that line's slope must be -4/3 (discussed in 1)

so, y=(-4/3)x+b ... Figure out b by substituting coordinates, (-3,8), as in problem 2.

 

4.-  same as problem 3, but parallel line has same slope as original line;  figure out slope of original line, by again getting the lequation into slope-intercept form, ie, get y=mx+b; the new line will have the same slope, m;

again, substitute given coordinates to determine value of the y-intercept, b.

 

5.-  figure out the slope, by (y2-y1)/(x2-x1) ... (5--5)/(2--7) = 10/9; then, using either of the coordinates given use the y-y1=slope*(x-x1) ... Y-5=(10/9)(x-2)