SOLUTION: Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions. 0.5x-2y=

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Question 80296This question is from textbook Algebra I
: Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.
0.5x-2y=17
2x+y=104 This question is from textbook Algebra I

Found 2 solutions by vertciel, jim_thompson5910:
Answer by vertciel(183) About Me  (Show Source):
You can put this solution on YOUR website!
Please note that we are not here to do your homework for you.
Show some work to prove that you have attempted at this problem please.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Multiply the 1st equation by 2 to get
x-4y=34
2x%2By=104
Now solve the system:

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


Lets start with the given system of linear equations

1%2Ax-4%2Ay=34
2%2Ax%2B1%2Ay=104

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

-4%2Ay=34-1%2AxSubtract 1%2Ax from both sides

y=%2834-1%2Ax%29%2F-4 Divide both sides by -4.


Which breaks down and reduces to



y=-17%2F2%2B%281%2F4%29%2Ax Now we've fully isolated y

Since y equals -17%2F2%2B%281%2F4%29%2Ax we can substitute the expression -17%2F2%2B%281%2F4%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


2%2Ax%2B1%2Ahighlight%28%28-17%2F2%2B%281%2F4%29%2Ax%29%29=104 Replace y with -17%2F2%2B%281%2F4%29%2Ax. Since this eliminates y, we can now solve for x.

2%2Ax%2B1%2A%28-17%2F2%29%2B1%281%2F4%29x=104 Distribute 1 to -17%2F2%2B%281%2F4%29%2Ax

2%2Ax-17%2F2%2B%281%2F4%29%2Ax=104 Multiply



2%2Ax-17%2F2%2B%281%2F4%29%2Ax=104 Reduce any fractions

2%2Ax%2B%281%2F4%29%2Ax=104%2B17%2F2Add 17%2F2 to both sides


2%2Ax%2B%281%2F4%29%2Ax=208%2F2%2B17%2F2 Make 104 into a fraction with a denominator of 2


2%2Ax%2B%281%2F4%29%2Ax=225%2F2 Combine the terms on the right side



%288%2F4%29%2Ax%2B%281%2F4%29x=225%2F2 Make 2 into a fraction with a denominator of 4

%289%2F4%29%2Ax=225%2F2 Now combine the terms on the left side.


cross%28%284%2F9%29%289%2F4%29%29x=%28225%2F2%29%284%2F9%29 Multiply both sides by 4%2F9. This will cancel out 9%2F4 and isolate x

So when we multiply 225%2F2 and 4%2F9 (and simplify) we get



x=50 <---------------------------------One answer

Now that we know that x=50, lets substitute that in for x to solve for y

2%2850%29%2B1%2Ay=104 Plug in x=50 into the 2nd equation

100%2B1%2Ay=104 Multiply

1%2Ay=104-100Subtract 100 from both sides

1%2Ay=4 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ay=%284%2F1%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=4%2F1 Multiply the terms on the right side


y=4 Reduce


So this is the other answer


y=4<---------------------------------Other answer


So our solution is

x=50 and y=4

which can also look like

(50,4)

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

1%2Ax-4%2Ay=34
2%2Ax%2B1%2Ay=104

we get


graph of 1%2Ax-4%2Ay=34 (red) and 2%2Ax%2B1%2Ay=104 (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 (50,4). This verifies our answer.


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

Plug in (50,4) into the system of equations


Let x=50 and y=4. Now plug those values into the equation 1%2Ax-4%2Ay=34

1%2A%2850%29-4%2A%284%29=34 Plug in x=50 and y=4


50-16=34 Multiply


34=34 Add


34=34 Reduce. Since this equation is true the solution works.


So the solution (50,4) satisfies 1%2Ax-4%2Ay=34



Let x=50 and y=4. Now plug those values into the equation 2%2Ax%2B1%2Ay=104

2%2A%2850%29%2B1%2A%284%29=104 Plug in x=50 and y=4


100%2B4=104 Multiply


104=104 Add


104=104 Reduce. Since this equation is true the solution works.


So the solution (50,4) satisfies 2%2Ax%2B1%2Ay=104


Since the solution (50,4) satisfies the system of equations


1%2Ax-4%2Ay=34
2%2Ax%2B1%2Ay=104


this verifies our answer.