SOLUTION: 2. For the following system: {2x+3y=8} {-3x+4y=-1} a. x value smaller than y b. y value smaller than x c. no solution d. infinite solutions

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: 2. For the following system: {2x+3y=8} {-3x+4y=-1} a. x value smaller than y b. y value smaller than x c. no solution d. infinite solutions      Log On


   

Question 104437: 2. For the following system:
{2x+3y=8}
{-3x+4y=-1}
a. x value smaller than y
b. y value smaller than x
c. no solution
d. infinite solutions
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

2%2Ax%2B3%2Ay=8
-3%2Ax%2B4%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

3%2Ay=8-2%2AxSubtract 2%2Ax from both sides

y=%288-2%2Ax%29%2F3 Divide both sides by 3.


Which breaks down and reduces to



y=8%2F3-%282%2F3%29%2Ax Now we've fully isolated y

Since y equals 8%2F3-%282%2F3%29%2Ax we can substitute the expression 8%2F3-%282%2F3%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


-3%2Ax%2B4%2Ahighlight%28%288%2F3-%282%2F3%29%2Ax%29%29=-1 Replace y with 8%2F3-%282%2F3%29%2Ax. Since this eliminates y, we can now solve for x.

-3%2Ax%2B4%2A%288%2F3%29%2B4%28-2%2F3%29x=-1 Distribute 4 to 8%2F3-%282%2F3%29%2Ax

-3%2Ax%2B32%2F3-%288%2F3%29%2Ax=-1 Multiply



-3%2Ax%2B32%2F3-%288%2F3%29%2Ax=-1 Reduce any fractions

-3%2Ax-%288%2F3%29%2Ax=-1-32%2F3 Subtract 32%2F3 from both sides


-3%2Ax-%288%2F3%29%2Ax=-3%2F3-32%2F3 Make -1 into a fraction with a denominator of 3


-3%2Ax-%288%2F3%29%2Ax=-35%2F3 Combine the terms on the right side



%28-9%2F3%29%2Ax-%288%2F3%29x=-35%2F3 Make -3 into a fraction with a denominator of 3

%28-17%2F3%29%2Ax=-35%2F3 Now combine the terms on the left side.


cross%28%283%2F-17%29%28-17%2F3%29%29x=%28-35%2F3%29%283%2F-17%29 Multiply both sides by 3%2F-17. This will cancel out -17%2F3 and isolate x

So when we multiply -35%2F3 and 3%2F-17 (and simplify) we get



x=35%2F17 <---------------------------------One answer

Now that we know that x=35%2F17, lets substitute that in for x to solve for y

-3%2835%2F17%29%2B4%2Ay=-1 Plug in x=35%2F17 into the 2nd equation

-105%2F17%2B4%2Ay=-1 Multiply

4%2Ay=-1%2B105%2F17Add 105%2F17 to both sides

4%2Ay=-17%2F17%2B105%2F17 Make -1 into a fraction with a denominator of 17



4%2Ay=88%2F17 Combine the terms on the right side

cross%28%281%2F4%29%284%29%29%2Ay=%2888%2F17%29%281%2F4%29 Multiply both sides by 1%2F4. This will cancel out 4 on the left side.

y=88%2F68 Multiply the terms on the right side


y=22%2F17 Reduce


So this is the other answer


y=22%2F17<---------------------------------Other answer


So our solution is

x=35%2F17 and y=22%2F17

which can also look like

(35%2F17,22%2F17)

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

2%2Ax%2B3%2Ay=8
-3%2Ax%2B4%2Ay=-1

we get


graph of 2%2Ax%2B3%2Ay=8 (red) and -3%2Ax%2B4%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 (35%2F17,22%2F17). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (35%2F17,22%2F17) into the system of equations


Let x=35%2F17 and y=22%2F17. Now plug those values into the equation 2%2Ax%2B3%2Ay=8

2%2A%2835%2F17%29%2B3%2A%2822%2F17%29=8 Plug in x=35%2F17 and y=22%2F17


70%2F17%2B66%2F17=8 Multiply


136%2F17=8 Add


8=8 Reduce. Since this equation is true the solution works.


So the solution (35%2F17,22%2F17) satisfies 2%2Ax%2B3%2Ay=8



Let x=35%2F17 and y=22%2F17. Now plug those values into the equation -3%2Ax%2B4%2Ay=-1

-3%2A%2835%2F17%29%2B4%2A%2822%2F17%29=-1 Plug in x=35%2F17 and y=22%2F17


-105%2F17%2B88%2F17=-1 Multiply


-17%2F17=-1 Add


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


So the solution (35%2F17,22%2F17) satisfies -3%2Ax%2B4%2Ay=-1


Since the solution (35%2F17,22%2F17) satisfies the system of equations


2%2Ax%2B3%2Ay=8
-3%2Ax%2B4%2Ay=-1


this verifies our answer.





Since we can see that the x-value for the solution is greater than the y-value, this means the answer is B