# Let’s find the equation of the line that passes through the point (4,-3) with a slope of -2

###### Question:

## Answers

1) first we have to find the slope of the linear function we have to find. As we know, it is perpendicular to the function 8x+7y-9=0. So that means that the slope of our function has to be opposite and inverse to the function we are given. To know the slope of the function we have to isolate Y.

8x+7y-9=0

7x=-8x+9

X=-8/7x+9/7

So the slope of this function is -8/7, which means that the slope of our function is 7/8.

So far we know that:

Y=7/8x+B

2) now we have to find the value of the intersection with Y (B). In order to do this we use the point (3;2) and supplant it on the equation.

2 = 7/8 . 3 + b

2 = 21/8 + b

2 - 21/8 = b

-5/8 = b

So the equation is:

Y = 7/8 x - 5/8

I hope you find my answer helpful

1) 7

2) -1

3) -4

4) y = 3x+9

5) y = x + 6

6) y = -¹/₃x + 10

7) g = 0.36

I have explanations for each of these problems with images at http://imgur.com/a/XviaD. I couldn't attach them because there are 7 questions and a maximum of 5 images.

1) 7 2) (-1,0) 3) m=-4 4) y=3x-4 5) y=x+6 6)

g(x)=-1/3x+10 7) y= $0.36

Step-by-step explanation:

These questions are all about Cartesian Geometry.

1) The Distance from point (7,-6) to vertical axis (0,-6) is measured with a straight line, between point (7,-6) and nearer point (0,-6)

d=|7-0|=7

2) To determine the value of k, let's determine the function with the two known points: (0, −1), (10, −11).

[tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\Rightarrow m=\frac{-11+1}{10-0}\Rightarrow m=-1\\-11=-1*(10)+b\therefore b=-1\Rightarrow f(x)=-x-1\\f(k)=-k-1\\0=-k-1\therefore k=-1[/tex]

So (k,0)=(-1,0)

3) To find the slope of the line, we must apply that formula used above.

[tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\Rightarrow m=\frac{7+9}{1-5}\Rightarrow m=\frac{16}{4}=-4[/tex]

4) To find the equation of the line which the midpoint (2,2) since Midpoint is given by

[tex]Midpoint=(\frac{x_{1}+x_{2}}{2}+\frac{y_{1}+y_{2}}{2})\\[/tex]

And the slope is 3, then m=3. Notice the formula is the same to calculate the slope, but we will only pick one point. Since (2,2) ∈ to the function let's use this point, as initial value (x0,y,0)

[tex]m(x-x_{0})=y-y_{0}\\3(x-2)=y-2\\3x-6=y-2\Rightarrow 3x-y=6-2\Rightarrow -y=4-3x\Rightarrow y=3x-4[/tex]

5) Similarly to the previous one:

[tex]m(x-x_{0})=y-y_{0}\\(x-0)=y-6\\x=y-6\Rightarrow -y=-x-6\Rightarrow y=x+6[/tex]

6) A ball being twirled. The center of the rotation is the origin of Coordinate System (0,0) when the string breaks at point (3,9)

If the line was straight from the origin to point (3,9)[tex]m=\frac{9-0}{3-0}\Rightarrow m=3\therefore 9=3(3)+b\Rightarrow b=0\Rightarrow f(x)=3x[/tex]

But the point (3,9) ∈ to a tangent line to the circumference described by the twirling.

Since it is perpendicular instead of m=3 it is -1/m i.e. -1/3, also the circumference intercepts the y-axis in ≈10

g(x)=-1/3x+10

7) In this case, we must also find the function.

x=units of electricity, y=units of gas |

500x+100y=331

400x+250y=326

The cost per unit of gas is y. Finding out the unit value for y

[tex]100y=331-500x\rightarrow y=\frac{331-500x}{100}\\500x=331-100y\\x=\frac{331-100y}{500}\\400(\frac{331-100y}{500})+250y=326\\52.96-80y+250y=326\Rightarrow y=\frac{9}{25} \,and \,x=\frac{59}{100}[/tex]

[tex]Iam about to fail this math class. i would like to verify that my answers are right. . find the dis[/tex]

1) 7 2) (-1,0) 3) m=-4 4) y=3x-4 5) y=x+6 6)

g(x)=-1/3x+10 7) y= $0.36

Step-by-step explanation:

These questions are all about Cartesian Geometry.

1) The Distance from point (7,-6) to vertical axis (0,-6) is measured with a straight line, between point (7,-6) and nearer point (0,-6)

d=|7-0|=7

2) To determine the value of k, let's determine the function with the two known points: (0, −1), (10, −11).

[tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\Rightarrow m=\frac{-11+1}{10-0}\Rightarrow m=-1\\-11=-1*(10)+b\therefore b=-1\Rightarrow f(x)=-x-1\\f(k)=-k-1\\0=-k-1\therefore k=-1[/tex]

So (k,0)=(-1,0)

3) To find the slope of the line, we must apply that formula used above.

[tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\Rightarrow m=\frac{7+9}{1-5}\Rightarrow m=\frac{16}{4}=-4[/tex]

4) To find the equation of the line which the midpoint (2,2) since Midpoint is given by

[tex]Midpoint=(\frac{x_{1}+x_{2}}{2}+\frac{y_{1}+y_{2}}{2})\\[/tex]

And the slope is 3, then m=3. Notice the formula is the same to calculate the slope, but we will only pick one point. Since (2,2) ∈ to the function let's use this point, as initial value (x0,y,0)

[tex]m(x-x_{0})=y-y_{0}\\3(x-2)=y-2\\3x-6=y-2\Rightarrow 3x-y=6-2\Rightarrow -y=4-3x\Rightarrow y=3x-4[/tex]

5) Similarly to the previous one:

[tex]m(x-x_{0})=y-y_{0}\\(x-0)=y-6\\x=y-6\Rightarrow -y=-x-6\Rightarrow y=x+6[/tex]

6) A ball being twirled. The center of the rotation is the origin of Coordinate System (0,0) when the string breaks at point (3,9)

If the line was straight from the origin to point (3,9)[tex]m=\frac{9-0}{3-0}\Rightarrow m=3\therefore 9=3(3)+b\Rightarrow b=0\Rightarrow f(x)=3x[/tex]

But the point (3,9) ∈ to a tangent line to the circumference described by the twirling.

Since it is perpendicular instead of m=3 it is -1/m i.e. -1/3, also the circumference intercepts the y-axis in ≈10

g(x)=-1/3x+10

7) In this case, we must also find the function.

x=units of electricity, y=units of gas |

500x+100y=331

400x+250y=326

The cost per unit of gas is y. Finding out the unit value for y

[tex]100y=331-500x\rightarrow y=\frac{331-500x}{100}\\500x=331-100y\\x=\frac{331-100y}{500}\\400(\frac{331-100y}{500})+250y=326\\52.96-80y+250y=326\Rightarrow y=\frac{9}{25} \,and \,x=\frac{59}{100}[/tex]

[tex]I am about to fail this math class. I would like to verify that my answers are right. Please help.[/tex]

[tex]y = \dfrac{-1}{4}x + \dfrac{-27}{4}[/tex]

Step-by-step explanation:

The equation of a line is:

y = mx + b

Where:

m = slope

b = y-intercept

First thing we need to do is solve for the slope. The slope formula is:

[tex]m = \dfrac{y_2-y_1}{x_2-x_1}[/tex]

Where:

x₁ = x-coordinate of the first point

x₂ = x-coordinate of the second point

y₁ = y-coordinate of the first point

y₂ = y-coordinate of the second point

We are given the following points:

Point 1: (-3, -6)

Point 2: (5, -8)

So let's plug in our coordinates into the slope formula:

[tex]m = \dfrac{y_2-y_1}{x_2-x_1}\\\\ =\dfrac{(-8)-(-6)}{5-(-3)}\\\\ =\dfrac{-2}{8}\\\\ =\dfrac{-1}{4}[/tex]

So we have our new equation of this line:

[tex]y = \dfrac{-1}{4}x + b[/tex]

What do we do then about the y-intercept?

Our points will help us out by plugging them in our equation, so we can solve for our y-intercept (b).

Let's do both to show that it would be the same:

Point 1 (-3, -6)

[tex]y = \dfrac{-1}{4}x + b\\\\-6 = \dfrac{-1}{4}(-3) + b\\\\-6 = \dfrac{3}{4} + b\\\\-6 = \dfrac{3}{4} + b\\\\subtract \dfrac{3}{4}\;from\;both\;sides\;of\;the\;equation\\\\-6- \dfrac{3}{4}= \dfrac{3}{4}- \dfrac{3}{4}+b\\\\\\\dfrac{-24-3}{4}=0+b\\\\\\-\dfrac{27}{4} = b[/tex]

Point 2: (5, -8)

[tex]y = \dfrac{-1}{4}x + b\\\\-8 = \dfrac{-1}{4}(5) + b\\\\-8 = \dfrac{-5}{4} + b\\\\-8 = \dfrac{-5}{4} + b\\\\subtract \dfrac{-5}{4}\;from\;both\;sides\;of\;the\;equation\\\\-8- \dfrac{-5}{4}= \dfrac{-5}{4}- \dfrac{-5}{4}+b\\\\\\\dfrac{-32-(-5)}{4}=0+b\\\\\\\dfrac{-27}{4} = b[/tex]

Now that we have b, we can insert that into the equation of the line:

[tex]y = \dfrac{-1}{4}x + b\\\\y = \dfrac{-1}{4}x + \dfrac{-27}{4}[/tex]

In standard form, your answer would be y = -0.5x - 2.5.

Step-by-step explanation:

If you put the points into the formula, (y2-y1)/(x2-x1), then you will be able to find the slope which is -0.5. Then, all that is needed is to find where the line that passes through both points passes through the y axis, which is -2.5.

The 2-point form of the equation for a line can be used.

... y = (y₂-y₁)/(x₂-x₁)·(x -x₁) +y₁

Filling in the given information, you have

... y = (4-1)/(4-3)·(x-3) +1 . . . . an equation for the line

... y = 3x -8 . . . . . . . . . . . . . . simplified to slope-intercept form

... 3x -y = 8 . . . . . . . . . . . . . .. rearranged to standard form

[tex]Find an equation of the line that passes through the given points. (let x be the independent variabl[/tex]

The required equation of line is [tex]7x-8y-5=0[/tex].

Step-by-step explanation:

Given : The point (3, 2) and is perpendicular to the line [tex]8x+7y-9 = 0[/tex].

To find : An equation of the line that passes through the point ?

Solution :

We know that,

When two lines are perpendicular then one slope is negative reciprocal of another slope.

The slope of [tex]8x+7y-9 = 0[/tex] line is

Write the equation in slope form,

[tex]7y=-8x+9[/tex]

[tex]y=-\frac{8}{7}x+\frac{9}{7}[/tex]

The slope is [tex]-\frac{8}{7}[/tex].

The slope of required equation is [tex]m=-(\frac{1}{-\frac{8}{7}})=\frac{7}{8}[/tex]

The point is [tex](x_1,y_1)=(3,2)[/tex].

The equation of required line is [tex]y-y_1=m(x-x_1)[/tex]

Substitute the value,

[tex]y-2=\frac{7}{8}(x-3)[/tex]

[tex]8(y-2)=7(x-3)[/tex]

[tex]8y-16=7x-21[/tex]

[tex]7x-8y-5=0[/tex]

Therefore, the required equation of line is [tex]7x-8y-5=0[/tex].

y = 2x + 4

Step-by-step explanation:

Find the slope m of the line passing through the 2 points using the slope formula

m = (y₂ - y₁ ) / (x₂ - x₁ )

with (x₁, y₁ ) = (- 2, - 3) and (x₂, y₂ ) = (2, 5)

m = [tex]\frac{5+3}{2+2}[/tex] = [tex]\frac{8}{4}[/tex] = 2

Parallel lines have equal slopes, thus

y = 2x + c ← is the partial equation of the line

To find c substitute← (- 1, 2) into the partial equation

2 = - 2 + c ⇒ c = 2 + 2 = 4

y = 2x + 4 ← is the equation of the line

Y = 2x + 5

By plugging the slope and coordinates into point slope form, you can get the answer in slope intercept form.