SOLUTION: choose the number solution to the following systems of equations. x+9y=9 3x-15y=-5

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Question 697463: choose the number solution to the following systems of equations.
x+9y=9
3x-15y=-5
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

by graphing
Solved by pluggable solver: Solve the System of Equations by Graphing



Start with the given system of equations:


1x%2B9y=9

3x-15y=-5





In order to graph these equations, we need to solve for y for each equation.




So let's solve for y on the first equation


1x%2B9y=9 Start with the given equation



9y=9-x Subtract +x from both sides



9y=-x%2B9 Rearrange the equation



y=%28-x%2B9%29%2F%289%29 Divide both sides by 9



y=%28-1%2F9%29x%2B%289%29%2F%289%29 Break up the fraction



y=%28-1%2F9%29x%2B1 Reduce



Now lets graph y=%28-1%2F9%29x%2B1 (note: if you need help with graphing, check out this solver)



+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+%28-1%2F9%29x%2B1%29+ Graph of y=%28-1%2F9%29x%2B1




So let's solve for y on the second equation


3x-15y=-5 Start with the given equation



-15y=-5-3x Subtract 3+x from both sides



-15y=-3x-5 Rearrange the equation



y=%28-3x-5%29%2F%28-15%29 Divide both sides by -15



y=%28-3%2F-15%29x%2B%28-5%29%2F%28-15%29 Break up the fraction



y=%281%2F5%29x%2B1%2F3 Reduce





Now lets add the graph of y=%281%2F5%29x%2B1%2F3 to our first plot to get:


Graph of y=%28-1%2F9%29x%2B1(red) and y=%281%2F5%29x%2B1%2F3(green)


From the graph, we can see that the two lines intersect at the point (15%2F7,16%2F21) (note: you might have to adjust the window to see the intersection)



or by substitution
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

1%2Ax%2B9%2Ay=9
3%2Ax-15%2Ay=-5

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

9%2Ay=9-1%2AxSubtract 1%2Ax from both sides

y=%289-1%2Ax%29%2F9 Divide both sides by 9.


Which breaks down and reduces to



y=1-%281%2F9%29%2Ax Now we've fully isolated y

Since y equals 1-%281%2F9%29%2Ax we can substitute the expression 1-%281%2F9%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


3%2Ax%2B-15%2Ahighlight%28%281-%281%2F9%29%2Ax%29%29=-5 Replace y with 1-%281%2F9%29%2Ax. Since this eliminates y, we can now solve for x.

3%2Ax-15%2A%281%29-15%28-1%2F9%29x=-5 Distribute -15 to 1-%281%2F9%29%2Ax

3%2Ax-15%2B%2815%2F9%29%2Ax=-5 Multiply



3%2Ax-15%2B%285%2F3%29%2Ax=-5 Reduce any fractions

3%2Ax%2B%285%2F3%29%2Ax=-5%2B15Add 15 to both sides


3%2Ax%2B%285%2F3%29%2Ax=10 Combine the terms on the right side



%289%2F3%29%2Ax%2B%285%2F3%29x=10 Make 3 into a fraction with a denominator of 3

%2814%2F3%29%2Ax=10 Now combine the terms on the left side.


cross%28%283%2F14%29%2814%2F3%29%29x=%2810%2F1%29%283%2F14%29 Multiply both sides by 3%2F14. This will cancel out 14%2F3 and isolate x

So when we multiply 10%2F1 and 3%2F14 (and simplify) we get



x=15%2F7 <---------------------------------One answer

Now that we know that x=15%2F7, lets substitute that in for x to solve for y

3%2815%2F7%29-15%2Ay=-5 Plug in x=15%2F7 into the 2nd equation

45%2F7-15%2Ay=-5 Multiply

-15%2Ay=-5-45%2F7Subtract 45%2F7 from both sides

-15%2Ay=-35%2F7-45%2F7 Make -5 into a fraction with a denominator of 7



-15%2Ay=-80%2F7 Combine the terms on the right side

cross%28%281%2F-15%29%28-15%29%29%2Ay=%28-80%2F7%29%281%2F-15%29 Multiply both sides by 1%2F-15. This will cancel out -15 on the left side.

y=-80%2F-105 Multiply the terms on the right side


y=16%2F21 Reduce


So this is the other answer


y=16%2F21<---------------------------------Other answer


So our solution is

x=15%2F7 and y=16%2F21

which can also look like

(15%2F7,16%2F21)

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

1%2Ax%2B9%2Ay=9
3%2Ax-15%2Ay=-5

we get


graph of 1%2Ax%2B9%2Ay=9 (red) and 3%2Ax-15%2Ay=-5 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (15%2F7,16%2F21). This verifies our answer.


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Check:

Plug in (15%2F7,16%2F21) into the system of equations


Let x=15%2F7 and y=16%2F21. Now plug those values into the equation 1%2Ax%2B9%2Ay=9

1%2A%2815%2F7%29%2B9%2A%2816%2F21%29=9 Plug in x=15%2F7 and y=16%2F21


15%2F7%2B144%2F21=9 Multiply


189%2F21=9 Add


9=9 Reduce. Since this equation is true the solution works.


So the solution (15%2F7,16%2F21) satisfies 1%2Ax%2B9%2Ay=9



Let x=15%2F7 and y=16%2F21. Now plug those values into the equation 3%2Ax-15%2Ay=-5

3%2A%2815%2F7%29-15%2A%2816%2F21%29=-5 Plug in x=15%2F7 and y=16%2F21


45%2F7-240%2F21=-5 Multiply


-105%2F21=-5 Add


-5=-5 Reduce. Since this equation is true the solution works.


So the solution (15%2F7,16%2F21) satisfies 3%2Ax-15%2Ay=-5


Since the solution (15%2F7,16%2F21) satisfies the system of equations


1%2Ax%2B9%2Ay=9
3%2Ax-15%2Ay=-5


this verifies our answer.