SOLUTION: Example: Find the point of intersection of the following lines. x= 32 + y 3x= 8y+ 6

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Question 473343: Example: Find the point of intersection of the following lines.
x= 32 + y
3x= 8y+ 6
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
x=+32+%2B+y
3x=+8y%2B+6
write both of them in standard form Ax%5E2%2BBy=C
x-y=32
3x-8y=6
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

1%2Ax-1%2Ay=32
3%2Ax-8%2Ay=6

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=32-1%2AxSubtract 1%2Ax from both sides

y=%2832-1%2Ax%29%2F-1 Divide both sides by -1.


Which breaks down and reduces to



y=-32%2B1%2Ax Now we've fully isolated y

Since y equals -32%2B1%2Ax we can substitute the expression -32%2B1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


3%2Ax%2B-8%2Ahighlight%28%28-32%2B1%2Ax%29%29=6 Replace y with -32%2B1%2Ax. Since this eliminates y, we can now solve for x.

3%2Ax-8%2A%28-32%29-8%281%29x=6 Distribute -8 to -32%2B1%2Ax

3%2Ax%2B256-8%2Ax=6 Multiply



3%2Ax%2B256-8%2Ax=6 Reduce any fractions

3%2Ax-8%2Ax=6-256 Subtract 256 from both sides


3%2Ax-8%2Ax=-250 Combine the terms on the right side



-5%2Ax=-250 Now combine the terms on the left side.


cross%28%281%2F-5%29%28-5%2F1%29%29x=%28-250%2F1%29%281%2F-5%29 Multiply both sides by 1%2F-5. This will cancel out -5%2F1 and isolate x

So when we multiply -250%2F1 and 1%2F-5 (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

3%2850%29-8%2Ay=6 Plug in x=50 into the 2nd equation

150-8%2Ay=6 Multiply

-8%2Ay=6-150Subtract 150 from both sides

-8%2Ay=-144 Combine the terms on the right side

cross%28%281%2F-8%29%28-8%29%29%2Ay=%28-144%2F1%29%281%2F-8%29 Multiply both sides by 1%2F-8. This will cancel out -8 on the left side.

y=-144%2F-8 Multiply the terms on the right side


y=18 Reduce


So this is the other answer


y=18<---------------------------------Other answer


So our solution is

x=50 and y=18

which can also look like

(50,18)

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

1%2Ax-1%2Ay=32
3%2Ax-8%2Ay=6

we get


graph of 1%2Ax-1%2Ay=32 (red) and 3%2Ax-8%2Ay=6 (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 (50,18). This verifies our answer.


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Check:

Plug in (50,18) into the system of equations


Let x=50 and y=18. Now plug those values into the equation 1%2Ax-1%2Ay=32

1%2A%2850%29-1%2A%2818%29=32 Plug in x=50 and y=18


50-18=32 Multiply


32=32 Add


32=32 Reduce. Since this equation is true the solution works.


So the solution (50,18) satisfies 1%2Ax-1%2Ay=32



Let x=50 and y=18. Now plug those values into the equation 3%2Ax-8%2Ay=6

3%2A%2850%29-8%2A%2818%29=6 Plug in x=50 and y=18


150-144=6 Multiply


6=6 Add


6=6 Reduce. Since this equation is true the solution works.


So the solution (50,18) satisfies 3%2Ax-8%2Ay=6


Since the solution (50,18) satisfies the system of equations


1%2Ax-1%2Ay=32
3%2Ax-8%2Ay=6


this verifies our answer.