Question

1. Write the equation in slope-intercept form. Identify the slope and y-intercept. SHOW ALL WORK

Answer 1

Part 1)

we know that

the equation of the line in slope-intercept form is equal to

$y=mx+b$

where

m is the slope

b is the y-intercept

we have

$2x-3y=9$

solve for y

$3y=2x-9$

$y=(2/3)x-3$ -------> equation of the line in slope-intercept form

so

the slope m  is $\frac{2}{3}$

the y-intercept b is $-3$

Part 2)

we know that

the equation of the line in slope-intercept form is equal to

$y=mx+b$

where

m is the slope

b is the y-intercept

we have

$x-4y=-20$

solve for y

$4y=x+20$

$y=(1/4)x+5$ -------> equation of the line in slope-intercept form

so

the slope m  is $\frac{1}{4}$

the y-intercept b is $5$

Part 3)

we know that

The x-intercept is the value of x when the value of y is equal to zero

The y-intercept is the value of y when the value of x is equal to zero

we have

$-x+4y=12$

a) Find the x-intercept

For $y=0$ substitute in the equation

$-x+4*0=12$

$x=-12$

The answer part 3a) is $(-12,0)$

b) Find the y-intercept

For $x=0$ substitute in the equation

$-0+4y=12$

$y=3$

The answer part 3b) is $(0,3)$

Part 4)

we know that

the equation of the line in standard form is

$Ax+By=C$  

we have

$y=\frac{2}{3}x+7$

Multiply by $3$ both sides

$3y=2x+21$

$2x-3y=-21$ ------> equation in standard form

therefore

the answer Part 4) is option B False

Part 5)

Step 1

Find the slope

we have

$2x-5y=12$

solve for y

$5y=2x-12$

$y=(2/5)x-(12/5)$

so

the slope m is $\frac{2}{5}$

Step 2

Find the y-intercept

The y-intercept is the value of y when the value of x is equal to zero

we have

$4y+24=5x$

for $x=0$

$4y+24=5*0$

$4y=-24$

$y=-6$

the y-intercept is $-6$

Step 3

Find the equation of the line

we have

$m=\frac{2}{5}$

$b=-6$

the equation of the line in slope-intercept form is

$y=mx+b$

substitute the values

$y=\frac{2}{5}x-6$

therefore

the answer Part 5) is the option A $y=\frac{2}{5}x-6$

Part 6)

Step 1

Find the slope of the given line

we know that

if two lines are perpendicular. then the product of their slopes is equal to minus one

so

$m1*m2=-1$

in this problem

the given line

$x+8y=27$

solve for y

$8y=27-x$

$y=(27/8)-(x/8)$

the slope m1 is $m1=-\frac{1}{8}$

so

the slope m2 is $m2=8$

Step 2

Find the equation of the line

we know that

the equation of the line in slope point form is equal to

$y-y1=m*(x-x1)$

we have

$m2=8$

point $(-5,5)$

substitutes the values

$y-5=8*(x+5)$

$y=8x+40+5$

$y=8x+45$

therefore

the answer part 6) is the option C $y=8x+45$

Part 7)

$y=(8/3)x+ 19$  -------> the slope is $m=(8/3)$

$8x- y=17$

$y =8x-17$ --------> the slope is $m=8$

we know that

if two lines are parallel , then their slopes are the same

in this problem the slopes are not the same

therefore

the answer part 7) is the option D) No, since the slopes are different.

Part 8)

a. Write an equation for the line in point-slope form

b. Rewrite the equation in standard form using integers

Step 1

Find the slope of the line

we know that

the slope between two points is equal to

$m=\frac{(y2-y1)}{(x2-x1)}$

substitute the values

$m=\frac{(4+1)}{(8-2)}$

$m=\frac{(5)}{(6)}$

Step 2

Find the equation in point slope form

we know that

the equation of the line in slope point form is equal to

$y-y1=m*(x-x1)$

we have

$m=(5/6)$

point $(2,-1)$

substitutes the values

$y+1=(5/6)*(x-2)$ -------> equation of the line in point slope form

Step 3

Rewrite the equation in standard form using integers

$y=(5/6)x-(5/3)-1$

$y=(5/6)x-(8/3)$

Multiply by $6$ both sides

$6y=5x-16$

$5x-6y=16$ --------> equation of the line in standard form

Part 9)

we know that

The formula to calculate the slope between two points is equal to

$m=\frac{(y2-y1)}{(x2-x1)}$

where

(x1,y1) ------> is the first point

(x2,y2) -----> is the second point

In the numerator calculate the difference of the y-coordinates

in the denominator calculate the difference of the x-coordinates

Part 10)

we know that

The formula to calculate the slope between two points is equal to

$m=\frac{(y2-y1)}{(x2-x1)}$

substitutes

$m=\frac{(5+1)}{(-1+3)}$

$m=\frac{(6)}{(2)}$

$m=3$

therefore

the answer Part 10) is $m=3$

Part 11)

we know that

the equation of the line in slope point form is equal to

$y-y1=m*(x-x1)$

substitute the values

$y+9=-2*(x-10)$ --------> this is the equation in the point slope form