Question

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2. Suppose y varies directly with x, and y = 15 and x = 5.
Write a direct variation equation that relates x and y.
What is the value of y when x = 9?

3. Write an equation in slope-intercept form of the line
that passed through (-3, 4) and (1. 4).

4. Use point-slope form to write the equation of a line
that has a slope of 2/3 and passes through (-3, -1).
Write your final equation in slope-intercept form.

5. Write the equation in standard form using integers
(no fractions or decimals): = −2/3 − 1

6. Write an equation of the line that passes through
(2, -1) and is parallel to the graph of y = 5x – 2. Write
your final equation in slope-intercept form.

Answer 1

When $y$ varies directly with $x$, the equation for the line will be $y=kx$ (called direct variation). It represents that $y$ is the result of $x$ being modified and nothing else being added or subtracted. Here's the math:

2. Suppose $y$ varies directly with $x$  and $y=15$ and $x=5$. Write a direct variation equation that relates $x$ and $y$. What is the value of $y$ when $x=9$? 

replace $x$ and $y$ with the given values in a direct variation equation

$y=kx$

$15=5k$

solve for $k$ (direct variation)

$k=3$

the equation for the line is $y=3x$, now solve for $y$ when $x=9$

$y=3(9)$

$y=27$

3. Write an equation in slope-intercept form of the line that passed through $(-3,4)$ and $(1,4)$.

write the expression for slope when $y=mx+b$

$m= \frac{ y_{2} - y_{1} }{x_2- x_{1} }$

define variables

$y_{2} =4 \\ y_{1} =4 \\ x_{2} =1 \\ x_{1} =-3$

insert values into slope expression

$m= \frac{ 4- 4}{1- (-3) }$

simplify

$m=0$

now, plug in the slope into the equation $y=mx+b$ 

$y=0x+b$

solve for $b$, plug in an $x$ and $y$ value from the given coordinate $(-3,4)$

$4=0(-3)+b$

$4=b$

$y=4$

4. Use point-slope form to write the equation of a line that has a slope of $2/3$ and passes through $(-3,-1)$. Write your final equation in slope-intercept form.

write the point slope form equation from the slope intercept form equation 

$y=mx+b \\ y- y_{1} =m(x- x_{1} )$

replace $y_{1}$ with $-1$ and $x_{1}$ with $-3$

$y- (-1) =m(x- (-3))$

simplify

$y+1 =m(x+3)$

insert the slope for $m$

$y+1 =2/3(x+3)$

simplify

$y+1 =2/3x+2$

$y=2/3x+3$

5. Write the equation in standard form using integers: $y=-2/3x-1$

standard form is $Ax+By=C$, so rearrange the variables into that form

$y=-2/3x-1$

$y+2/3x=-1$

multiply both sides by $3$

$3y+2x=-3$

6. Write an equation of the line that passes through $(2,-1)$ and is parallel to the graph of $y=5x-2$. Write your final equation in slope-intercept form. 

parallel lines have the same slope, but different y-intercepts

to solve for the y-intercept of the second line, replace x and y with given values

$-1 = 5(2) + b$

$-1 = 10 + b$

$b=-11$

$y=5x-11$

ANSWERS:

2. $y=3x$ and $y=27$ when $x=9$

3. $y=4$

4. $y=2/3x+3$

5. $3y+2x=-3$

6. $y=5x-11$

Hope this helps! I have added the answer to Question #5!

Answer 2

2. Suppose y varies directly with x, and y = 15 and x = 5. Write a direct variation equation that relates x and y. What is the value of y when x = 9? 

replace x and y with the given values in a direct variation equation

y = kx

15 = 5k

solve for k (direct variation)

k = 3

the equation for the line is y = 3x, now solve for y when x = 9

y = 3(9)

y = 27

3. Write an equation in slope-intercept form of the line that passed through (-3, 4) and (1, 4).

write the expression for slope when y = mx + b



define variables



insert values into slope expression



simplify



now, plug in the slope into the equation y = mx + b 



solve for b, plug in an x and y value from the given coordinate (-3,4)







4. Use point-slope form to write the equation of a line that has a slope of 2/3 and passes through (-3, -1). Write your final equation in slope-intercept form.

write the point slope form equation from the slope intercept form equation 



replace  with -1 and  with -3



simplify



insert the slope for m



simplify





5. Write the equation in standard form using integers: = −2/3 − 1

standard form is Ax+By=C, so I need the actual equation in order to give you the answer...

6. Write an equation of the line that passes through (2, -1) and is parallel to the graph of y = 5x – 2. Write your final equation in slope-intercept form. 

parallel lines have the same slope, but different y-intercepts

to solve for the y-intercept of the second line, replace x and y with given values









ANSWERS:

2.  and  when