SOLUTION: 3x+y=13 x+6y=-7

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Question 87248: 3x+y=13
x+6y=-7
Answer by jim_thompson5910(35256) About Me  (Show Source):
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If you want to solve the system by substitution (you need to provide more details), then...

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


Lets start with the given system of linear equations

3%2Ax%2B1%2Ay=13
1%2Ax%2B6%2Ay=-7

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=13-3%2AxSubtract 3%2Ax from both sides

y=%2813-3%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=13-3%2Ax Now we've fully isolated y

Since y equals 13-3%2Ax we can substitute the expression 13-3%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B6%2Ahighlight%28%2813-3%2Ax%29%29=-7 Replace y with 13-3%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax%2B6%2A%2813%29%2B6%28-3%29x=-7 Distribute 6 to 13-3%2Ax

1%2Ax%2B78-18%2Ax=-7 Multiply



1%2Ax%2B78-18%2Ax=-7 Reduce any fractions

1%2Ax-18%2Ax=-7-78 Subtract 78 from both sides


1%2Ax-18%2Ax=-85 Combine the terms on the right side



-17%2Ax=-85 Now combine the terms on the left side.


cross%28%281%2F-17%29%28-17%2F1%29%29x=%28-85%2F1%29%281%2F-17%29 Multiply both sides by 1%2F-17. This will cancel out -17%2F1 and isolate x

So when we multiply -85%2F1 and 1%2F-17 (and simplify) we get



x=5 <---------------------------------One answer

Now that we know that x=5, lets substitute that in for x to solve for y

1%285%29%2B6%2Ay=-7 Plug in x=5 into the 2nd equation

5%2B6%2Ay=-7 Multiply

6%2Ay=-7-5Subtract 5 from both sides

6%2Ay=-12 Combine the terms on the right side

cross%28%281%2F6%29%286%29%29%2Ay=%28-12%2F1%29%281%2F6%29 Multiply both sides by 1%2F6. This will cancel out 6 on the left side.

y=-12%2F6 Multiply the terms on the right side


y=-2 Reduce


So this is the other answer


y=-2<---------------------------------Other answer


So our solution is

x=5 and y=-2

which can also look like

(5,-2)

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

3%2Ax%2B1%2Ay=13
1%2Ax%2B6%2Ay=-7

we get


graph of 3%2Ax%2B1%2Ay=13 (red) and 1%2Ax%2B6%2Ay=-7 (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 (5,-2). This verifies our answer.


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Check:

Plug in (5,-2) into the system of equations


Let x=5 and y=-2. Now plug those values into the equation 3%2Ax%2B1%2Ay=13

3%2A%285%29%2B1%2A%28-2%29=13 Plug in x=5 and y=-2


15-2=13 Multiply


13=13 Add


13=13 Reduce. Since this equation is true the solution works.


So the solution (5,-2) satisfies 3%2Ax%2B1%2Ay=13



Let x=5 and y=-2. Now plug those values into the equation 1%2Ax%2B6%2Ay=-7

1%2A%285%29%2B6%2A%28-2%29=-7 Plug in x=5 and y=-2


5-12=-7 Multiply


-7=-7 Add


-7=-7 Reduce. Since this equation is true the solution works.


So the solution (5,-2) satisfies 1%2Ax%2B6%2Ay=-7


Since the solution (5,-2) satisfies the system of equations


3%2Ax%2B1%2Ay=13
1%2Ax%2B6%2Ay=-7


this verifies our answer.