SOLUTION: Solve the system using any method. (Simplify your answer completely. If the system is dependent, enter a general solution in terms of x and y. If there is no solution, enter NO SOL

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: Solve the system using any method. (Simplify your answer completely. If the system is dependent, enter a general solution in terms of x and y. If there is no solution, enter NO SOL      Log On


   

Question 1130656: Solve the system using any method. (Simplify your answer completely. If the system is dependent, enter a general solution in terms of x and y. If there is no solution, enter NO SOLUTION.)
4x + 5y = 2
16x − 15y = 1
Found 2 solutions by MathLover1, MathTherapy:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

4x+%2B+5y+=+2
16x+-15y+=+1

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


Lets start with the given system of linear equations

4%2Ax%2B5%2Ay=2
16%2Ax-15%2Ay=1

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

5%2Ay=2-4%2AxSubtract 4%2Ax from both sides

y=%282-4%2Ax%29%2F5 Divide both sides by 5.


Which breaks down and reduces to



y=2%2F5-%284%2F5%29%2Ax Now we've fully isolated y

Since y equals 2%2F5-%284%2F5%29%2Ax we can substitute the expression 2%2F5-%284%2F5%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


16%2Ax%2B-15%2Ahighlight%28%282%2F5-%284%2F5%29%2Ax%29%29=1 Replace y with 2%2F5-%284%2F5%29%2Ax. Since this eliminates y, we can now solve for x.

16%2Ax-15%2A%282%2F5%29-15%28-4%2F5%29x=1 Distribute -15 to 2%2F5-%284%2F5%29%2Ax

16%2Ax-30%2F5%2B%2860%2F5%29%2Ax=1 Multiply



16%2Ax-6%2B12%2Ax=1 Reduce any fractions

16%2Ax%2B12%2Ax=1%2B6Add 6 to both sides


16%2Ax%2B12%2Ax=7 Combine the terms on the right side



28%2Ax=7 Now combine the terms on the left side.


cross%28%281%2F28%29%2828%2F1%29%29x=%287%2F1%29%281%2F28%29 Multiply both sides by 1%2F28. This will cancel out 28%2F1 and isolate x

So when we multiply 7%2F1 and 1%2F28 (and simplify) we get



x=1%2F4 <---------------------------------One answer

Now that we know that x=1%2F4, lets substitute that in for x to solve for y

16%281%2F4%29-15%2Ay=1 Plug in x=1%2F4 into the 2nd equation

4-15%2Ay=1 Multiply

-15%2Ay=1-4Subtract 4 from both sides

-15%2Ay=-3 Combine the terms on the right side

cross%28%281%2F-15%29%28-15%29%29%2Ay=%28-3%2F1%29%281%2F-15%29 Multiply both sides by 1%2F-15. This will cancel out -15 on the left side.

y=-3%2F-15 Multiply the terms on the right side


y=1%2F5 Reduce


So this is the other answer


y=1%2F5<---------------------------------Other answer


So our solution is

x=1%2F4 and y=1%2F5

which can also look like

(1%2F4,1%2F5)

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

4%2Ax%2B5%2Ay=2
16%2Ax-15%2Ay=1

we get


graph of 4%2Ax%2B5%2Ay=2 (red) and 16%2Ax-15%2Ay=1 (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 (1%2F4,1%2F5). This verifies our answer.


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Check:

Plug in (1%2F4,1%2F5) into the system of equations


Let x=1%2F4 and y=1%2F5. Now plug those values into the equation 4%2Ax%2B5%2Ay=2

4%2A%281%2F4%29%2B5%2A%281%2F5%29=2 Plug in x=1%2F4 and y=1%2F5


4%2F4%2B5%2F5=2 Multiply


40%2F20=2 Add


2=2 Reduce. Since this equation is true the solution works.


So the solution (1%2F4,1%2F5) satisfies 4%2Ax%2B5%2Ay=2



Let x=1%2F4 and y=1%2F5. Now plug those values into the equation 16%2Ax-15%2Ay=1

16%2A%281%2F4%29-15%2A%281%2F5%29=1 Plug in x=1%2F4 and y=1%2F5


16%2F4-15%2F5=1 Multiply


20%2F20=1 Add


1=1 Reduce. Since this equation is true the solution works.


So the solution (1%2F4,1%2F5) satisfies 16%2Ax-15%2Ay=1


Since the solution (1%2F4,1%2F5) satisfies the system of equations


4%2Ax%2B5%2Ay=2
16%2Ax-15%2Ay=1


this verifies our answer.

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

Solve the system using any method. (Simplify your answer completely. If the system is dependent, enter a general solution in terms of x and y. If there is no solution, enter NO SOLUTION.)
4x + 5y = 2
16x − 15y = 1
How come you haven't learned how to do these problems YET? There might have been more than 30 of these over the past few days.

Why have you chosen someone or some people on here to do your math assignments for you? All these problems are the same! Why can't you

now do them by yourself after being shown how some are done?



If you decide to do this one (considering the fact that the other person who's been solving these problems has chosen to use the most

inefficient and most time-consuming method), then ALL you need to do is the following:

1)  Multiply eq (i) by 3 to form eq (iii) and then add eqs (iii) & (ii). This eliminates y and gives you the value of x

2)  Substitute the value of x in any of the 2 ORIGINAL equations to find y.

That's IT. NOTHING more, NOTHING less!!