Question

SOMEONE PLEASE HELP ME!! I'll give a medal!
The graph below shows a system of equations:


The x-coordinate of the solution to the system of equations is __________.

Answer 1

https://gbwebacademy.brainhoney.com/Resource/22373151,A49,0,38/Assets/73504_53bff066/module5_graph24.gif

Answer 2

You know that the solution to the system of equations is the point where all the lines intersect, right?

Answer 3

Uhm no haha. Math is my worst subject, I am surprised I even moved on to Algebra.

Answer 4

@whpalmer4

Answer 5

Wait so it's 1?

Answer 6

Never mind! I was looking at the wrong graph!!

Answer 7

Okay, well, each of the straight lines in the graph represents the set of solutions to one of the equation. The solution to a system of equations is going to be where those lines cross, because they cross at a point they share, right?

Answer 8

Is the graph you posted the link to the correct graph?

Answer 9

Well yes, they meet at 2 right? And yes that's the correct graph

Answer 10

Either 2 or -2?

Answer 11

the x coordinate (the value on the x-axis) is the the right-left value, and the y coordinate (the value on the y-axis) is the up-down value. given that, which do you think is correct?

Answer 12

and I can guarantee that the person who made this problem chose to have one be 2 and the other be -2 just to confuse you (well, to see how well you understand, really)!

Answer 13

So 2? Because it says x-coordinate

Answer 14

-2 I meant

Answer 15

okay, that's better :-)

Answer 16

Okay thank you! Could you help me with a few other please??

Answer 17

others*

Answer 18

lol — do you find many guys that say no to you? :-)

Answer 19

Well a lot have lol, I hope this means you'll help?:)

Answer 20

of course, that's why I'm here instead of fixing the dishwasher :-)

Answer 21

you'll fix my dishwasher in exchange, right? :-)

Answer 22

If you need to do that I can wait haha it's okay, as long as you come back! lol

Answer 23

I can't guarantee I'll fix it but all I can do is try right? :)

Answer 24

okay, let's see the next problem. my goal is to be so successful in showing you how to do them that you won't need to ask many!

Answer 25

Sally bought x pounds of apples and y pounds of pears. Three times the number of pounds of pears she bought was 3 more than the number of pounds of apples she bought. The total number of pounds of apples and pears she bought was 5 pounds. Which graph best shows the number of pounds of apples and pears Sally bought?

https://gbwebacademy.brainhoney.com/Resource/22373151,A49,0,38/Assets/73504_53bff066/module5_graph85.gif
https://gbwebacademy.brainhoney.com/Resource/22373151,A49,0,38/Assets/73504_53bff066/module5_graph89.gif
https://gbwebacademy.brainhoney.com/Resource/22373151,A49,0,38/Assets/73504_53bff066/module5_graph86.gif
https://gbwebacademy.brainhoney.com/Resource/22373151,A49,0,38/Assets/73504_53bff066/module5_graph87.gif

Answer 26

Okay, this one is a bit trickier. You'll need to write some equations to represent what the words in the problem say.

Sally bought x pounds of apples and y pounds of pears. [...] The total number of pounds of apples and pears she bought was 5 pounds.

Can you write an equation that says that?

Answer 27

x+y=5?

Answer 28

excellent!

Answer 29

"Three times the number of pounds of pears she bought was 3 more than the number of pounds of apples she bought."

Now, how about this one?

Answer 30

Well this one is confusing me a little bit... Would this make sense for pears? 3x=y
Probably not but I am trying..

Answer 31

well, just go through the sentence replacing the words with the variable names
apples are x
pears are y

so

three times y is 3 more than x

what is that in equation form?

Answer 32

"three times y" becomes \(3y\)
"is" becomes \(=\)
"3 more than x" becomes \(3+x\)

so that gives us...

Answer 33

3y=3+x
Sorry I was getting there, I know I am slow/:

Answer 34

is it suppose to be 3y=x+3 or is it fine the way it is?

Answer 35

Okay, so we have two equations, and that means we have two lines (in theory, they could end up being the same line, but I can tell by looking that they are not, and so can you!)

As we said before, the solution of the system of equations will be the point where all of the lines intersect.

\[3y=x+3\]is just as good as \[x+3 = 3y\]or\[x = 3y-3\]or any of the other arrangements

Answer 36

Now we need to figure out which of those graphs correspond to our equations. One way would be to plug a few numbers into the equation, and see if the results appear on the graph we are considering. I always like to try 0 because it usually makes the work easy. So let's plug 0 into both of our equations for \(x\) and see what pops out for \(y\). When we do so, we'll have a pair of points, and we can look at each graph and discard any graph which does not have both of those points on the lines.

Answer 37

\[x+y=5\]well \(x=0\) so \[0+y=5\]means that \[y=5\]so our first point is \((0,5)\)
any question about what I did there?

Answer 38

Uhm, I don't think so. I understand.

Answer 39

So far...

Answer 40

okay, can you plug \(x=0\) in to the other equation and find out the corresponding value for \(y\)?

Answer 41

Well where are you getting the equations? From the graphs?

Answer 42

no, we just figured them out from the story problem, remember? the other equation is \[3y = 3+x\]

Answer 43

now, if we were serious math geeks, we could look at the graphs and say "oh, the equation for this line is such-and-such" and work from that direction, but we aren't, so we won't :-)

Answer 44

I don't want to look stupid and give the wrong answer... /:

Answer 45

nah, you won't look stupid. if it looks like you gave the wrong answer, I'll avert my eyes quickly :-)

Answer 46

so we substitute \(0\) for \(x\) in the equation \[3y = 3+x\]which gives us \[3y = 3 + 0\] or \[3y = 3\]

Answer 47

y+3=x
x=0
so y+3=0
so y=3?
Don't laugh.. I'm sure i look really dumb. I was just looking at what you did for the first one..

Answer 48

mmm...not quite. our first equation was \[x+y = 5\]and after I replaced \(x\) with \(0\) that became \[0+y=5\]\[y=5\]so our first point is \((0,5)\)

Our second equation was \[3y=3+x\] and after I replaced \(x\) with \(0\) that became \[3y = 3 + 0 \]\[3y=3\]divide both sides by the coefficient of \(y\) and we get\[\frac{3y}{3} = \frac{3}{3}\]\[y = 1\]so our second point is \((0,1)\)

clear as mud?

Answer 49

Oh okay, well that seemed really easy. Now I feel stupid...

Answer 50

Now, which of the graphs have both \((0,5)\) and \((0,1)\) as points on the lines?

Answer 51

Yes, taking \(x=0\) makes the arithmetic easy most of the time. Sometimes \(x=1\) is also easy, if you need to repeat the process at a different spot. \(x=539.427634\) is usually NOT easy :-)

Answer 52

the "easy" values are also easy to find on the graph

Answer 53

Graph b?

Answer 54

If graph b is this one, then yes:

Answer 55

Yes it is :)

Answer 56

As a check, notice that the lines intersect at a point - can you tell me the x-coordinate of that point?

Answer 57

3?

Answer 58

Very good. To double-check our choice, let's figure out from the equations what the values of \(y\) should be at \(x=3\). If we did everything correctly, the values will be the same as the \(y\) coordinate of that point where they intersect.

Answer 59

Wow, I didn't realize how easily you can figure this stuff out. Thank you so much! Mind helping with a few more? If you have the time...

Answer 60

There's a saying in medical school: "see one, do one, teach one"

You saw how I found the \(y\) coordinates for \(x=0\)
Now you get to find the \(y\) coordinates for \(x=3\)
Then you'll be all set to teach a friend :-)

Answer 61

\[x+y=5\]\[3+y=5\]\[3-3+y=5-3\]\[y=2\]and\[3y=3+x\]\[3y=3+3\]\[3y=6\]\[y=2\]so both of our equations supply \((3,2)\) as a point on the graph, and our chosen graph does have \((3,2)\) as a point of intersection, so we did it correctly.

Next problem?

Answer 62

The graph of the following system of equations is
6x + 3y = 12


A. overlapping lines
B. parallel lines
C. intersecting lines

Answer 63

and 2x + y = 6

Answer 64

@whpalmer4

Answer 65

okay, our equations are \[6x+3y=12\]\[2x+y=6\]

do you notice anything about the coefficients of the two equations?

Answer 66

Uhm no not necessarily..

Answer 67

sorry, had an interruption

compare the coefficients of the \(x\)'s
now compare the coefficients of the \(y\)'s
see a pattern?

Answer 68

oh no worries, and i think so.

Answer 69

the coefficient in the first equation is 3x the coefficient in the second equation, right?

Answer 70

Wait would the second one be 6?

Answer 71

if we have an equation of a line, say

\[ax + by = c\]where \(a,b,c\) are constant values and \(x,y\) are variables, we can multiply each term of the equation by the same value and the resulting equation draws exactly the same line.

For example, \[x+y=5\] from our previous problem. If we multiply the equation by 2, we get \[2x + 2y = 10\]

Let's find out what the points would be on that line at \(x=0\) and \(x=3\):

\[2(0)+2y=10\]\[0+2y=10\]\[2y=10\]\[y=5\]so \((0,5)\) is a point again
\[2(3)+2y=10\]\[6+2y=10\]\[2y=4\]\[y=2\]so \((3,2)\) is also a point again

If the equation of one line is a multiple of the equation of another, the two lines are identical, or coincident.

Answer 72

Hmm, this is getting confusing.

Answer 73

However, if they have different values for the constant term, but the coefficients are multiples like this, then they are parallel.

\[x+y=5\]\[x+y=1\]are parallel lines, because for any given value of \(x\), the \(y\) value for one is going to be exactly \(5-1=4\) larger than the \(y\) value for the other

Answer 74

some pictures might help! can you wait a few minutes, I have to do something for She Who Must Be Obeyed

Answer 75

Haha yes that's completely fine with me

Answer 76

Here is a plot of \[x+y=5\]

Answer 77

and here is a plot of \[x+y=1\]

Answer 78

Here is a plot of both of them on the same piece of paper:

Answer 79

Those are parallel lines, agreed?

Answer 80

Agreed

Answer 81

Because they are parallel, they will not intersect. For any value of \(x\) we plug into \[x+y=1\]the value of \(y\) we get will be 4 less than the value of \(y\) we get plugging the same value of \(x\) into \[x+y=5\]

Answer 82

Now I was talking about multiplying an equation by a constant not affecting the line at all. Let's do a picture of that.

Here is \[x+y=5\] again

Answer 83

Okay I am slowly understanding

Answer 84

Here is \[2x+2y=10\]

Answer 85

Exactly the same!

Answer 86

Oh yeah it is the same

Answer 87

So, if we can make one equation into an exactly copy of the other equation by just multiplying (or dividing) by a number, the two lines are coincident (or overlapping as your answer key puts it).

We can't do that with this problem because while multiplying

\[3*2x+3*y=3*6\]
gives us \[6x+3y\]on the left hand side of the equals sign, which is exactly what our other equation has, the value on the right hand is still different. That means that the lines are parallel, but do not overlap.

The third case is if the lines intersect in a point. For this to happen, the coefficients cannot be a multiple of each other. If they are not, somewhere on the infinite plane that contains the two lines, they will intersect.

If you arrange the equation of a line in the form
\[Ax+By=C\] where \(A,B,C\) are all constants (numbers), you can pretty quickly judge which of the three cases you have.

If multiplying one equation by any number will give you matching values of \(A,B,C\), the two lines are overlapping. This means there are infinitely many solutions that satisfy the system of equations.

If multiplying one equation by any number will give you matching values of \(A,B\) but not \(C\), then the lines are parallel. This means there are 0 solutions that satisfy the system of equations.

Finally, if you can't multiply by any number to get a match of \(A,B\) for the two equations, that means the lines intersect. There will be 1 solution that satisfies the system of equations, and that is the poin

Answer 88

is the point of intersection.

Answer 89

I'm sure your eyes must have glazed over by now, right? :-)

Answer 90

That's just a lot to read and understand but I think if I read it over closely I will understand but from where I am looking, they don't overlap but they are parallel? but you said something is the point of intersection but I am not sure what you were talking about

Answer 91

Okay, quick recap: our equations were

\[6x+3y=12\]\[
2x+y=6\]

If we multiply the second equation by 3, the coefficients for the \(x\) and \(y\) terms match each other:

\[6x+3y=12\]\[
3*2x+3*y=3*6\]

\[6x+3y=12\]\[
6x+3y=18\]

but the constant on the other side of the equals sign is different. That means we have parallel lines. By my classification scheme, we have matching \(A,B\) but not \(C\)

Answer 92

If we plot them both, here's what we get:

Answer 93

So they are parallel

Answer 94

Yep, that's right!

Answer 95

We covered more ground than necessary to solve the problem, but the concepts are important, and my hope was that getting a good handle on them will help you with many problems.