SOLUTION: 3x-y=-3 2x+y=-7 i am trying to figure out how to find the x,y,xy corrdinates so i can graph the equation to find out this problem has one solution, no solution, or infininate s

Algebra ->  Linear-equations -> SOLUTION: 3x-y=-3 2x+y=-7 i am trying to figure out how to find the x,y,xy corrdinates so i can graph the equation to find out this problem has one solution, no solution, or infininate s      Log On


   

Question 199610This question is from textbook intermediate algebra
: 3x-y=-3
2x+y=-7
i am trying to figure out how to find the x,y,xy corrdinates so i can graph the equation to find out this problem has one solution, no solution, or infininate solutions This question is from textbook intermediate algebra

Found 2 solutions by jim_thompson5910, vleith:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Start with the given system of equations:


system%283x-y=-3%2C2x%2By=-7%29


In order to graph these equations, we must solve for y first.


Let's graph the first equation:


3x-y=-3 Start with the first equation.


-y=-3-3x Subtract 3x from both sides.


y=%28-3-3x%29%2F%28-1%29 Divide both sides by -1 to isolate y.


y=3x%2B3 Rearrange the terms and simplify.


Looking at y=3x%2B3 we can see that the equation is in slope-intercept form y=mx%2Bb where the slope is m=3 and the y-intercept is b=3


Since b=3 this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis

So we have one point




Now since the slope is comprised of the "rise" over the "run" this means
slope=rise%2Frun

Also, because the slope is 3, this means:

rise%2Frun=3%2F1


which shows us that the rise is 3 and the run is 1. This means that to go from point to point, we can go up 3 and over 1



So starting at , go up 3 units


and to the right 1 unit to get to the next point



Now draw a line through these points to graph y=3x%2B3

So this is the graph of y=3x%2B3 through the points and



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Now let's graph the second equation:


2x%2By=-7 Start with the second equation.


y=-7-2x Subtract 2x from both sides.


y=-2x-7 Rearrange the terms and simplify.


Looking at y=-2x-7 we can see that the equation is in slope-intercept form y=mx%2Bb where the slope is m=-2 and the y-intercept is b=-7


Since b=-7 this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis

So we have one point




Now since the slope is comprised of the "rise" over the "run" this means
slope=rise%2Frun

Also, because the slope is -2, this means:

rise%2Frun=-2%2F1


which shows us that the rise is -2 and the run is 1. This means that to go from point to point, we can go down 2 and over 1



So starting at , go down 2 units


and to the right 1 unit to get to the next point



Now draw a line through these points to graph y=-2x-7

So this is the graph of y=-2x-7 through the points and


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Now let's graph the two equations together:


Graph of y=3x%2B3 (red). Graph of y=-2x-7 (green)


From the graph, we can see that the two lines intersect at the point . So the solution to the system of equations is .

This means that the system has one unique solution.

This also tells us that the system of equations is consistent and independent.

Answer by vleith(2983) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

3%2Ax-1%2Ay=-3
2%2Ax%2B1%2Ay=-7

In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).

So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 3 and 2 to some equal number, we could try to get them to the LCM.

Since the LCM of 3 and 2 is 6, we need to multiply both sides of the top equation by 2 and multiply both sides of the bottom equation by -3 like this:

2%2A%283%2Ax-1%2Ay%29=%28-3%29%2A2 Multiply the top equation (both sides) by 2
-3%2A%282%2Ax%2B1%2Ay%29=%28-7%29%2A-3 Multiply the bottom equation (both sides) by -3


So after multiplying we get this:
6%2Ax-2%2Ay=-6
-6%2Ax-3%2Ay=21

Notice how 6 and -6 add to zero (ie 6%2B-6=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%286%2Ax-6%2Ax%29-2%2Ay-3%2Ay%29=-6%2B21

%286-6%29%2Ax-2-3%29y=-6%2B21

cross%286%2B-6%29%2Ax%2B%28-2-3%29%2Ay=-6%2B21 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

-5%2Ay=15

y=15%2F-5 Divide both sides by -5 to solve for y



y=-3 Reduce


Now plug this answer into the top equation 3%2Ax-1%2Ay=-3 to solve for x

3%2Ax-1%28-3%29=-3 Plug in y=-3


3%2Ax%2B3=-3 Multiply



3%2Ax=-3-3 Subtract 3 from both sides

3%2Ax=-6 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ax=%28-6%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.


x=-2 Multiply the terms on the right side


So our answer is

x=-2, y=-3

which also looks like

(-2, -3)

Notice if we graph the equations (if you need help with graphing, check out this solver)

3%2Ax-1%2Ay=-3
2%2Ax%2B1%2Ay=-7

we get



graph of 3%2Ax-1%2Ay=-3 (red) 2%2Ax%2B1%2Ay=-7 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (-2,-3). This verifies our answer.