# How do you find the slope of a line that passes through (5,2) and (-3,6)

## Answers

1. For the data in the table, does y vary directly with X? If it does write an equation for the direct variation.(X,y) (8,11) (16,22) (24,33)

Yes y=1.375x

2.for the data in the table, does y vary directly with X? If it does write an equation for the direct variation. (X,y) (16,4) (32,16) (48,36)

No y does not very directly with x***

3. (Time/hour,distance/miles)(4,233) (6,348) (8,464) (10, 580)

Express the relationship between distance and time in a simplified form as a unit rate. Determine which statement correctly interprets this relationship.

58/1 your car travels 58 miles in 1 hour

4.what is the slope of the line that passes through the pair of points (2,5) and (8,3)

-1/3

4.what is the slope of the line that passes through the pair of points (-5.2,8.7) and (-3.2,2.7)

-3

5. What is the slope of the line that passes through the pair of points (3/2,-2) and (-3,7/3)

-26/27

6.write an equation in point slope from for the line through the given point with the given slope (5,2) m=3

Y-2=3(X-5)

7. Write an equation in point slope form for the line through the given point with the given slope (-3,-5) m=-2/5

Y+5=-2/5(X+3)

8. Write an equation in point slope from for the line through the given point with the given slope. (4,-7) m=-0.54

Y+7=-0.54(x-4)

9. The table shows the height of a plant as it grows. Which equation in point slope from gives the plants height (time,plant height) (2,16)(4,32)(6,48)(8,64)

Y-16=8(X-2)***

10. Write y=-2/3x+7 in standard form

2x+3y=21

11. Write y=-1/2x+1 in standard form using integers

X+2y=2

[tex]\bf ~~~~~~~~~~~~\textit{distance between 2 points}\\\\ \begin{array}{ccccccccc} &&x_1&&y_1&&x_2&&y_2\\ % (a,b) &C&(~ -3 &,& 1~) % (c,d) &D&(~ 5 &,& 6~) \end{array}~ % distance value d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ CD=\sqrt{[5-(-3)]^2+[6-1]^2}\implies CD=\sqrt{(5+3)^2+(6-1)^2} \\\\\\ CD=\sqrt{8^2+5^2}\implies CD=\sqrt{64+25}\implies CD=\sqrt{89}[/tex]

[tex]\bf ~~~~~~~~~~~~\textit{middle point of 2 points }\\\\ \begin{array}{ccccccccc} &&x_1&&y_1&&x_2&&y_2\\ % (a,b) &F&(~ -6 &,& 4~) % (c,d) &G&(~ 8 &,& -2~) \end{array}\qquad % coordinates of midpoint \left(\cfrac{ x_2 + x_1}{2}\quad ,\quad \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left( \cfrac{8-6}{2}~~,~~\cfrac{-2+4}{2} \right)\implies \left( \cfrac{2}{2}~~,~~\cfrac{2}{2} \right)\implies (1,1)[/tex]

[tex]\bf \begin{array}{ccccccccc} &&x_1&&y_1&&x_2&&y_2\\ % (a,b) &&(~ -2 &,& -3~) % (c,d) &&(~ 1 &,& 1~) \end{array} \\\\\\ % slope = m slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{1-(-3)}{1-(-2)}\implies \cfrac{1+3}{1+2}\implies \cfrac{4}{3}[/tex]

1. y varies directly with x and the equation is [tex]y=1.375x[/tex]

2. No, y does not vary directly with x

3. Your car travels 58 miles in 1 hour

4. [tex]-\frac{1}{3}[/tex]

4. [tex]-3[/tex]

5. [tex]-\frac{26}{27}[/tex]

6. [tex]y-2=3(x-5)[/tex]

7. [tex]y+5=-\frac{2}{5}(x+3)[/tex]

8. [tex]y+7=-0.54(x-4)[/tex]

9. [tex]y-16=8(x-2)[/tex]

10. [tex]2x+3y=21[/tex]

11. [tex]x+2y=2[/tex]

Step-by-step explanation:

1.

For [tex]y[/tex] to vary directly with [tex]x[/tex] , all the 3 pair of numbers need to show the same ratio if we divide each y's by the x's. Let's check.

[tex]\frac{11}{8}=1.375[/tex][tex]\frac{22}{16}=1.375[/tex][tex]\frac{33}{24}=1.375[/tex]So all of them show the same ratio and hence y varies directly with x.

For equation, we already saw that multiplying x by 1.375 gives us y. We can write in equation form as:

[tex]y=1.375x[/tex]

Third answer choice is correct.

2.

This is similar to #1. So let's check the ratios.

[tex]\frac{4}{16}=0.25[/tex][tex]\frac{16}{32}=0.5[/tex][tex]\frac{36}{48}=0.75[/tex]As we can see, the ratios are not equal to y does not vary directly with x.

Fourth answer choice is correct.

3.

The first number in the pair gives time and second number gives distance. To get unit rate, we divide the distance by time. So we will get the number of miles traveled in 1 hour.

[tex]\frac{233}{4}=58.25[/tex] [ i believe this is a typo and it should be 232 miles and ratio would be 58 ]

[tex]\frac{348}{6}=58[/tex]

[tex]\frac{464}{8}=58[/tex]

[tex]\frac{580}{10}=58[/tex]

As we can see, in 1 hour, distance covered is 58 miles. Third answer choice is right.

4.

If the 2 points are taken as [tex](x_{1},y_{1})[/tex] and [tex](x_{2},y_{2})[/tex]

And we know formula of slope to be:

[tex]\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]

The slope of the line is:

[tex]\frac{3-5}{8-2}=\frac{-2}{6}=-\frac{1}{3}[/tex]

The slope of the line is [tex]-\frac{1}{3}[/tex]

Second answer choice is correct.

4. [this should be #5]

If the 2 points are taken as [tex](x_{1},y_{1})[/tex] and [tex](x_{2},y_{2})[/tex]

And we know formula of slope to be:

[tex]\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]

The slope of this line can be found now:

[tex]\frac{2.7-8.7}{-3.2-(-5.2)}=\frac{2.7-8.7}{-3.2+5.2}=\frac{-6}{2}=-3[/tex]

The slope of the line is [tex]-3[/tex]

Fourth answer choice is correct.

5.

The formula of slope is:

[tex]\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]

Where,

the 2 points are taken as [tex](x_{1},y_{1})[/tex] and [tex](x_{2},y_{2})[/tex]

Now finding the slope:

[tex]\frac{\frac{7}{3}-(-2)}{-3-\frac{3}{2}}=\frac{\frac{7}{3}+2}{-\frac{9}{2}}=\frac{\frac{13}{3}}{-\frac{9}{2}}=-\frac{26}{27}[/tex]

The slope of the line is [tex]-\frac{26}{27}[/tex]

Second answer choice is right.

6.

Point Slope form of a line is given as:

[tex]y-y_{1}=m(x-x_{1})[/tex]

Where,

[tex](x_{1},y_{1})[/tex] is the point given, andm is the slopeUsing the point (5, 2) and slope as 3 given, we can write the equation:

[tex]y-2=3(x-5)[/tex]

Fourth answer choice is right.

7.

Point Slope form of a line is given as:

[tex]y-y_{1}=m(x-x_{1})[/tex]

Where,

[tex](x_{1},y_{1})[/tex] is the point given, andm is the slopeUsing the point given as [tex](-3,-5)[/tex] and slope as [tex]m=-\frac{2}{5}[/tex] , we can write the point slope form of the equation as:

[tex]y-(-5)=-\frac{2}{5}(x-(-3))\\y+5=-\frac{2}{5}(x+3)[/tex]

First answer choice is right.

8.

Point Slope form of a line is given as:

[tex]y-y_{1}=m(x-x_{1})[/tex]

Where,

[tex](x_{1},y_{1})[/tex] is the point given, andm is the slopeThe slope is given as [tex]-0.54[/tex] and the point is (4, -7). So the point slope form is:

[tex]y-(-7)=-0.54(x-4)\\y+7=-0.54(x-4)[/tex]

First answer choice is right.

9.

In this question, we can just have a quick look and see that the [tex]y[/tex]-coordinate is 8 times the [tex]x[/tex]-coordinate. So we can say that [tex]y=8x[/tex]

Expanding the equations below would tell us which one is equal to that. Let's check.

[tex]y-16=8(x-2)\\y-16=8x-16\\y=8x-16+16\\y=8x[/tex]

This is the correct one.

So first answer choice is right.

10.

The standard form of the equation of a line is given as:

[tex]Ax+By=C[/tex]

Rearranging the given equation gives us:

[tex]y=-\frac{2}{3}x+7\\\frac{2}{3}x+y=7[/tex]

Now, we can't have a fraction, so we multiply all of it by 3 to get rid of the denominator. Now we have:

[tex]3*(\frac{2}{3}x+y=7)\\2x+3y=21[/tex]

First answer choice is right.

11.

The standard form of a line is [tex]Ax+By=C[/tex]

Rearranging the given equation, we have:

[tex]y=-\frac{1}{2}x+1\\\frac{1}{2}x+y=1[/tex]

We cannot have fractions, so we multiply the whole thing by 2 to get rid of the denominator. So we have:

[tex]2*(\frac{1}{2}x+y=1)\\x+2y=2[/tex]

First answer choice is correct.

2 and 9 are correct. The remainder need to be reconsidered.

Step-by-step explanation:

Your answer to 9 is correct, which means you have the appropriate signs for the point coordinates. In your other point-slope questions, one or more of the signs is incorrect. Use your good work as an example of how to choose the correct answers elsewhere.

1) When you have y = kx, you can find the value of k from ...

... k = y/x

When the variation is direct, this works for any (x, y) pair.

3) A unit rate is called a unit rate because it has 1 (a unit) in the denominator.

4a, 4b, ... all slope calculations. You managed to get it right in problem 9. Watch your signs.

10, 11 ... It can work well to put the x term on the same side with the y-term, then multiply the equation by the denominator value.

... (2/3)x + y = 7 . . . . add (2/3)x to both sides of the equation

... 2x + 3y = 21 . . . . . multiply by 3. (Standard form always has the leading coefficient positive.)

Whenever you consider making a term disappear from one side of the equation, think in terms of adding its opposite to both sides of the equation. This will give the result you want and is consistent with the properties of equality: whatever you do to one side of the equation you must also do to the other side of the equation. Other wording may have been used by your teacher or friends. This is the one true statement about transforming equations.

Use photo math they give a graph for each