SOLUTION: OK, the problem is: Solve the system of equations using the substitution method. If the answer is a unique solution, present it as an ordered pair: (x, y). If not, specify whether

Algebra ->  Linear-equations -> SOLUTION: OK, the problem is: Solve the system of equations using the substitution method. If the answer is a unique solution, present it as an ordered pair: (x, y). If not, specify whether      Log On


   

Question 82152: OK, the problem is: Solve the system of equations using the substitution method.
If the answer is a unique solution, present it as an ordered pair: (x, y). If not, specify whether the answer is “no solution” or “infinitely many solutions.”
-7x + y = 4
2x + 8y = 0
my understanding is; I should first solve the top equation because the y has a coefficient of 1 (not sure why, but that's my guess) when I do this I get:
x = -(-y+4/7) and then I am lost . . .
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

-7%2Ax%2B1%2Ay=4
2%2Ax%2B8%2Ay=0

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

1%2Ay=4%2B7%2AxAdd 7%2Ax to both sides

y=%284%2B7%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=4%2B7%2Ax Now we've fully isolated y

Since y equals 4%2B7%2Ax we can substitute the expression 4%2B7%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


2%2Ax%2B8%2Ahighlight%28%284%2B7%2Ax%29%29=0 Replace y with 4%2B7%2Ax. Since this eliminates y, we can now solve for x.

2%2Ax%2B8%2A%284%29%2B8%287%29x=0 Distribute 8 to 4%2B7%2Ax

2%2Ax%2B32%2B56%2Ax=0 Multiply



2%2Ax%2B32%2B56%2Ax=0 Reduce any fractions

2%2Ax%2B56%2Ax=0-32 Subtract 32 from both sides


2%2Ax%2B56%2Ax=-32 Combine the terms on the right side



58%2Ax=-32 Now combine the terms on the left side.


cross%28%281%2F58%29%2858%2F1%29%29x=%28-32%2F1%29%281%2F58%29 Multiply both sides by 1%2F58. This will cancel out 58%2F1 and isolate x

So when we multiply -32%2F1 and 1%2F58 (and simplify) we get



x=-16%2F29 <---------------------------------One answer

Now that we know that x=-16%2F29, lets substitute that in for x to solve for y

2%28-16%2F29%29%2B8%2Ay=0 Plug in x=-16%2F29 into the 2nd equation

-32%2F29%2B8%2Ay=0 Multiply

8%2Ay=0%2B32%2F29Add 32%2F29 to both sides

8%2Ay=0%2F29%2B32%2F29 Make 0 into a fraction with a denominator of 29



8%2Ay=32%2F29 Combine the terms on the right side

cross%28%281%2F8%29%288%29%29%2Ay=%2832%2F29%29%281%2F8%29 Multiply both sides by 1%2F8. This will cancel out 8 on the left side.

y=32%2F232 Multiply the terms on the right side


y=4%2F29 Reduce


So this is the other answer


y=4%2F29<---------------------------------Other answer


So our solution is

x=-16%2F29 and y=4%2F29

which can also look like

(-16%2F29,4%2F29)

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

-7%2Ax%2B1%2Ay=4
2%2Ax%2B8%2Ay=0

we get


graph of -7%2Ax%2B1%2Ay=4 (red) and 2%2Ax%2B8%2Ay=0 (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 (-16%2F29,4%2F29). This verifies our answer.


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Check:

Plug in (-16%2F29,4%2F29) into the system of equations


Let x=-16%2F29 and y=4%2F29. Now plug those values into the equation -7%2Ax%2B1%2Ay=4

-7%2A%28-16%2F29%29%2B1%2A%284%2F29%29=4 Plug in x=-16%2F29 and y=4%2F29


112%2F29%2B4%2F29=4 Multiply


116%2F29=4 Add


4=4 Reduce. Since this equation is true the solution works.


So the solution (-16%2F29,4%2F29) satisfies -7%2Ax%2B1%2Ay=4



Let x=-16%2F29 and y=4%2F29. Now plug those values into the equation 2%2Ax%2B8%2Ay=0

2%2A%28-16%2F29%29%2B8%2A%284%2F29%29=0 Plug in x=-16%2F29 and y=4%2F29


-32%2F29%2B32%2F29=0 Multiply


0%2F29=0 Add


0=0 Reduce. Since this equation is true the solution works.


So the solution (-16%2F29,4%2F29) satisfies 2%2Ax%2B8%2Ay=0


Since the solution (-16%2F29,4%2F29) satisfies the system of equations


-7%2Ax%2B1%2Ay=4
2%2Ax%2B8%2Ay=0


this verifies our answer.