Questions on Algebra: Linear Equations, Graphs, Slope answered by real tutors!

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x + y =5
2x - y =4
.
x=5-y
2(5-y)-y=4 Replace x with 5-y.
10-2y-y=4
-3y=-6
y=2
x=3
.
Ed

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Hello,
First we want to get it so we can cancel either the x or y when we add both equations together. So let's multiply the first equation by -4 to give us:
-20x+8y=+32 Now we can add this to the second equation.
3x-8y=-32
We get -17x=0 so x = 0
Now plug 0 in for x in either original equation and we get:
5(0)-2y=-8
y=4
To check these put them into both equations and it should prove correct.
RJ Toftness
www.math-unlock.com

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(-5,1) and (4,6)
Slope=(6-1)/(4+5)
Slope=5/9 ans.
1=5/9*-5+b
1=-25/9+b
b=1+25/9
b=9/9+25/9
b=34/9 ans. for the y intercept.
Y=5x/9+34/9
(graph 300x200 pixels, x from -6 to 5, y from -10 to 10, 5x/9 +34/9).

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(1/8,1/9) and (3/8,2/9)
Slope=(y2-y1)/(x2-x1)
Slope=(2/9-1/9)/(3/8-1/8)
Slope=(1/9)/(2/8)
Slope=(1/9)*(8/2) (invert the denominator & multiply)
Slope=8/18
Slope=4/9 ans.

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I have a LOT of resources on my own website on the topic of GRAPHING and SLOPE of a LINE. To find my website, do a "BING" or a "GOOGLE" search for my last name "Rapalje". Near the top of the search list, you should see "Rapalje Homepage." Click on this, and look near the top of my homepage for the link "Basic, Intermediate and College Algebra: One Step at a Time." Choose "Basic Algebra", and look in "Chapter 4" for "Section 4.02, Slope Intercept Method." This is my own explanation especially written for students who have trouble understanding math. I wrote this to students, NOT mathematicians, and I think you will find it easier to understand than your own traditional textbooks!! In addition to this page, the entire website is supported by my "MATH IN LIVING COLOR" pages, where the hardest problems are solved IN COLOR.

In addition, I have a video of me teaching this topic in the classroom a few years ago. To see this FREE video, from my Homepage, look for "Rapalje Videos in Living Color". Select "Basic Algebra", and look for "Graphing and Slope of a Line."

If you like my website, please recommend it to family and friends who have trouble understanding math!! It's ALL FREE!!!

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f(x)=e^(2-x)
----------
Pick values for x and find f(x).
x....f(x)
2.....1
1.....e
0.....e^2 etc.
--------------
Plot the points and draw a curve thru them.
Use Excel to do the calculations, or get the FREE software at
http://www.padowan.dk.com

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1. Graph the equations by completing the tables and plotting the points. You may use Excel or another web-based graphing utility.
a). y = -½ x + 3
Use the table; find at least 3 points using any values for x.
Sub each x and find y
x....y
-2...4 --> (-2,4)
0....3 --> (0,3)
4....1 --> (4,1)
Plot the points and draw a straight line thru them.
------------------
b). Use the values of x provided in the table.
Is this the same equation?
x y
0 3
1 2.5
2 2
3 1.5
4 1
----------

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1. A line passes through (2, –1) and (8, 4).
Find the slope using the slope formula: m =
Assign these as follows
x1=2; y1=-1
x2=8; y2=4
:
m = = = is the slope
:
a. Write an equation for the line in point-slope form.
The point-slope form: y - y1 = m(x - x1)
y - (-1) = (x - 2)
y + 1 = x - *2
y + 1 = x -
y = x - -1
y = x - -
:
y = x - ; is the point-slope form
:
;
b. Rewrite the equation in standard form using integers.
y = x -
Multiply each term by 6 to get rid of the denominators, results
6y = 5x - 2(8)
6y = 5x - 16
-5x + 6y = - 16
They prefer the 1s term to be positive, multiply by -1
5x - 6y = 16
;
:
You can check our equation:
substitute 8 for x and 4 for y and ensure they = 16

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x + y > 2....This one has no solution because we need to know the x or y value.
===============================================================================

y < 1
<------------0-------1----------->
Shade everything starting at 1 going in the left direction.
===========================================================


x > 0
<-----------0------------>
Shade everything starting at 0 and going in the right direction.

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This takes too long to type.
After doing the math on paper, I got x = -4 and y = 3.
The solution is the point (-4, 3). This is the point where the two lines meet on the xy-plane.

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(2,9); x=8y=7 YOU HAVE ONE TOO MANY = SIGNS.
CORRECT & RESUBMIT.

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Hello,
Let's add both equations.
x+y=7
x-y=9 It will look like this:
2x=16 (Notice the y cancels out)Now divide both sides by 2
x=8 Plug 8 into the original equation (either one)
8+y=7 Subtract 8 from both sides gives us:
y=-1 There we go. You can check these by plugging in the answers into the equations.
RJ Toftness
www.math-unlock.com

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Write an equation in slope-intercept form for the line that is parallel to the given line and that passes through the given point.
4y = x; (4,0)
Put equation in slope intercept form y - mx + b
4y = x
divide both sides by 4
y = x
the slope is 1/4 and the intercept is 0
;
Parallel lines have equal slopes:
:
Use the point slope formula: y - y1 = mx(x - x1)
In this problem m = 1/4, x1 = 4; y1 = 0
:
y - 0 = (x - 4)
y = x - 1 is line parallel to 4y=x and go thru 4,0

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15x + 9y = –27
9y=-15x-27
y=-15x/9-27/9
y=-5x/3-3 ans. (red line)
Slope=-5/3 y intercept=-3
------------------------
–7x + 6y = 84
6y=7x+84
y=7x/6+84/6
y=7x/6+14 ans. (green line)
x intercept=-12, y intercept=14
-----------------------------------
(graph 300x200 pixels, x from -12 to 5, y from -10 to 10, of TWO functions -5x/3 -3 and 7x/6 +14).

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5x-2y=10
-2y=-5x+10
y=-5x/-2+10/-2
y=5x/2-5 (red line) this line has a slope=5/2
A parallel line will also have a slope=5/2
(3,-5)
-5=5/2*3+b
-5=15/2+b
b=-5-15/2
b=-10/2-15/2
b=-25/2
Y=5X/2-25/2 (green line)ANS. FOR THE LINE PARALLEL & PASSING THROUGH (3,-5)
(graph 300x200 pixels, x from -6 to 6, y from -15 to 10, of TWO functions 5x/2 +5,5x/2 -25/2).

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(-2,5) & (5,4) ARE 2 POINTS.
Y=mX+b IS THE LINE EQUATION WERE m=slope & b= the Y intercept.
Find the slope(m):
m=(Y2-Y1)/(X2-X1)=(4-5)/(5+2)=-1/7
Using one set of X & Y values & the slope solve the line equation for b the Y intercept.
5=-1/7*-2+b
5=2/7+b
b=5-2/7=35/7-2/7=33/7 the Y intercept.
Y=-X/7+33/7 THE LINE EQUATION.
(graph 300x200 pixels, x from -6 to 5, y from -10 to 10, -x/7 +33/7).

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A line described by set equal to a constant is a horizontal line with slope of 0. Any line parallel must have the same slope, and therefore also be a horizontal line. If the line is horizontal and passes through (2, 4), then all of the ordered pairs that are in the solution set of the equation must have a coordinate equal to 4.


John

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The given equation is already in slope-intercept form, so you can determine the slope of the given line by inspection of the coefficient on .

Use the fact that perpendicular lines have slopes that are negative reciprocals of each other, that is:



to determine the slope of the desired line.

Now use the point-slope form of the equation of a line to write an equation of the desired line, that is:



where is the given point and is the slope you determined in the previous step.

You should ask your instructor or check for any instructions in your text as to the required final form of your answer such as slope-intercept or standard form.

Unfortunately, I cannot answer your question exactly as you stated it. One cannot find the equation of a line. That is because for any given set of ordered pairs that describe a straight line in a plane, there are an infinite number of equations that describe the relationship. The best anyone can ever do is provide an equation of a line.

John

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find the slope. through (4,-2) and parallel to x=1

That's a vertical line, and it has NO slope!

All equations of the form 

x = a

are vertical lines which are " a " units from the y-axis.

The vertical line whose equation is x = 1 is this green
line below:



The point (4,-2) is this point plotted below:



The line parallel to the vertical line whose
equation is x = 1 which goes through the point
(4,-2) is the blue vertical line below:



Its equation is x = 4.  But neither line has a slope.
Vertical lines are the only kind of lines which have no slope.
All other lines have slopes.

Edwin

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Look at this tool --> http://www.algebra.com/algebra/homework/Linear-equations/equation-parallel-or-perpendicular.solver

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In the following exercise, find the coordinates of the vertex for the parabola defined by the given quadratic function.
f(x)=4x^2+8x+1
-----------
The vertex occurs where x = -b/2a = -8/(2*4) = -1
---
f(-1) = 4(-1)^2 + 8(-1) + 1
f(-1) = 4-8+1
f(-1) = -3
-------------------
Vertex: (-1,-3)
======================
Cheers,
Stan H.

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To put this in slope intercept form, you have to solve for y. Start by subtracting 5x from each side.

5x-7y=9

-7y=-5x+9

Now divide by -7:



Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

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We use the point-slope formula.
y - y1 = m(x - x1)
y - 1 = 6(x - 1)
y - 1 = 6x - 6
y = 6x - 6 + 1
y = 6x - 5
==============================
By the way, the point-slope formula can be better understood as
y - y coordinate of your point = slope(x - x coordinate of your point)

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SLOPE=(-2+7)/(-9+5)=5/-4=-5/4 ANS.
SLOPE=(-12+17)/-1-3)=5/-4=-5/4 ANS.
SLOPE=(-7+12)/(-5+1)=5/-4=-5/4 ANS.
THIS PROVES THAT THESE POINTS FORM A STRAIGHT LINE WITH A SLOPE=-5/4.
Y=mX+b
-2=-5/4*-9+b
-2=45/4+b
-2=11.25+b
b=-2-11.25
b=-13.25 ans. for the y intercept.
Y=-5X/2-13.25 LINE EQUATION.
(graph 300x200 pixels, x from -12 to 5, y from -15 to 10, -5x/4 -13.25) 12

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9/3=3
15/5=3
21/7=3
27/9=3
THUS 3 IS THE MONTHLY MULTIPLIER.
12*3=36 IS THE PREDICTED HEIGHT OF THE PLANT AFTER 12 MONTHS.

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(-2,4) and (- 1, 1).
Y=mX+b
M=(Y2-Y1)/(X2-X1)=(1-4)/(-1+2)=-3/1=-3 ANS. FOR THE SLOPE.
4=-3*-2+b
4=6+b
B=4-6
b=-2 ANS. FOR THE Y INTERCEPT.
Y=-3X-2
(graph 300x200 pixels, x from -6 to 5, y from -10 to 10, -3x -2).
ANSWER. D) (-2/3,0)

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(6,12) and (0,-6)
Y=mX+b
m=(Y2-Y1)/(X2-X1)=(-6-12)/(0-6)=-18/-6=3 ans. for the slope.
-6=3*0+b
-6=b the Y intercept.
Y=3X-6
(graph 300x200 pixels, x from -6 to 5, y from -10 to 10, 3x -6).
ANSWER: A(3,3)

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Hello,
Let's start by showing x for the 4th test. So now we can set up and equation.
(x+87+81+88)/4=87 OK, now let's multiply by 4 to remove the fraction.
x+87=81+88=348 Combine like terms
x+256=348 Subtract 256 from both sides to get:
x=92
There we go. Needs a score of 92 to get and average of 87.
Plug it in to see if it works.
RJ Toftness
www.math-unlock.com

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Hello,
Okay, here we go. First "linear" means having to do with a line so this equation can be graphed to look like a straight line.
Another way to write this is
50-x-1(3x+2)=0 Now we can remove the parenthesis by multiplying -1.
50-x-3x-2=0 Now combine like terms.
48-4x=0 Now subtract 48 from both sides gives us:
-4x=-48 Now divide -4 into both sides gives us
x=12 There you go. Make sense for you now?
You might want to get my book it explains the basics easily.
RJ Toftness
www.math-unlock.com

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x is 0 secord y is 0, the third x is 1, the fourth was y is 3.
3x+2y=6
3*0+2Y=6
0+2Y=6
2Y=6
Y=6/2
Y=3 ANS. (0,3)
3X+2*0=6
3X+0=6
X=6/3
X=2 ANS. (2,0)
3*1+2Y=6
3+2Y=6
2Y=6-3
2Y=3
Y=3/2 ANS. (3/2,2)
3X+2*3=6
3X+6=6
3X=6-6
3X=0
X=0 ANS. (0,3)

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the table is of x and y pairs that make the equation true
you pick a value for x, and then find the y value

if x is 0, then y must be 1

if x is -4, then y must be 5

and so on...

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What data point? What line? What is the slope? You have to give us something to work with.

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80=10X
X=80/10
X=8 MG OF VITAMIN C IN AN APPLE.

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y=-4x+3