SOLUTION: solve by substituion 2x-3y+6z=-21 -5x+4y+z=3 7x-7y-4z=-6

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Question 166357: solve by substituion
2x-3y+6z=-21
-5x+4y+z=3
7x-7y-4z=-6
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
-5x%2B4y%2Bz=3 Start with the second equation


z=3%2B5x-4y Get every term but "z" to the right side

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2x-3y%2B6z=-21 Move onto the first equation


2x-3y%2B6%283%2B5x-4y%29=-21 Plug in z=3%2B5x-4y


2x-3y%2B18%2B30x-24y=-21 Distribute


2x-3y%2B30x-24y=-21-18 Subtract 18 from both sides.


32x-27y=-39 Combine like terms. So let's call this equation 4

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7x-7y-4z=-6 Move onto the third equation


7x-7y-4%283%2B5x-4y%29=-6 Plug in z=3%2B5x-4y


7x-7y-12-20x%2B16y=-6 Distribute


7x-7y-20x%2B16y=-6%2B12 Add 12 to both sides


-13x%2B9y=6 Combine like terms. So let's call this equation 5.


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So we now have the equations

32x-27y=-39 Equation 4
-13x%2B9y=6 Equation 5


Now let's solve the given system of equations 4 and 5


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


Lets start with the given system of linear equations

32%2Ax-27%2Ay=-39
-13%2Ax%2B9%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

-27%2Ay=-39-32%2AxSubtract 32%2Ax from both sides

y=%28-39-32%2Ax%29%2F-27 Divide both sides by -27.


Which breaks down and reduces to



y=13%2F9%2B%2832%2F27%29%2Ax Now we've fully isolated y

Since y equals 13%2F9%2B%2832%2F27%29%2Ax we can substitute the expression 13%2F9%2B%2832%2F27%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


-13%2Ax%2B9%2Ahighlight%28%2813%2F9%2B%2832%2F27%29%2Ax%29%29=6 Replace y with 13%2F9%2B%2832%2F27%29%2Ax. Since this eliminates y, we can now solve for x.

-13%2Ax%2B9%2A%2813%2F9%29%2B9%2832%2F27%29x=6 Distribute 9 to 13%2F9%2B%2832%2F27%29%2Ax

-13%2Ax%2B117%2F9%2B%28288%2F27%29%2Ax=6 Multiply



-13%2Ax%2B13%2B%2832%2F3%29%2Ax=6 Reduce any fractions

-13%2Ax%2B%2832%2F3%29%2Ax=6-13 Subtract 13 from both sides


-13%2Ax%2B%2832%2F3%29%2Ax=-7 Combine the terms on the right side



%28-39%2F3%29%2Ax%2B%2832%2F3%29x=-7 Make -13 into a fraction with a denominator of 3

%28-7%2F3%29%2Ax=-7 Now combine the terms on the left side.


cross%28%283%2F-7%29%28-7%2F3%29%29x=%28-7%2F1%29%283%2F-7%29 Multiply both sides by 3%2F-7. This will cancel out -7%2F3 and isolate x

So when we multiply -7%2F1 and 3%2F-7 (and simplify) we get



x=3 <---------------------------------One answer

Now that we know that x=3, lets substitute that in for x to solve for y

-13%283%29%2B9%2Ay=6 Plug in x=3 into the 2nd equation

-39%2B9%2Ay=6 Multiply

9%2Ay=6%2B39Add 39 to both sides

9%2Ay=45 Combine the terms on the right side

cross%28%281%2F9%29%289%29%29%2Ay=%2845%2F1%29%281%2F9%29 Multiply both sides by 1%2F9. This will cancel out 9 on the left side.

y=45%2F9 Multiply the terms on the right side


y=5 Reduce


So this is the other answer


y=5<---------------------------------Other answer


So our solution is

x=3 and y=5

which can also look like

(3,5)

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

32%2Ax-27%2Ay=-39
-13%2Ax%2B9%2Ay=6

we get


graph of 32%2Ax-27%2Ay=-39 (red) and -13%2Ax%2B9%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 (3,5). This verifies our answer.


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Check:

Plug in (3,5) into the system of equations


Let x=3 and y=5. Now plug those values into the equation 32%2Ax-27%2Ay=-39

32%2A%283%29-27%2A%285%29=-39 Plug in x=3 and y=5


96-135=-39 Multiply


-39=-39 Add


-39=-39 Reduce. Since this equation is true the solution works.


So the solution (3,5) satisfies 32%2Ax-27%2Ay=-39



Let x=3 and y=5. Now plug those values into the equation -13%2Ax%2B9%2Ay=6

-13%2A%283%29%2B9%2A%285%29=6 Plug in x=3 and y=5


-39%2B45=6 Multiply


6=6 Add


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


So the solution (3,5) satisfies -13%2Ax%2B9%2Ay=6


Since the solution (3,5) satisfies the system of equations


32%2Ax-27%2Ay=-39
-13%2Ax%2B9%2Ay=6


this verifies our answer.






z=3%2B5x-4y Go back to the first isolated equation


z=3%2B5%283%29-4%285%29 Plug in x=3 and y=5


z=3%2B15-20 Multiply.


z=-2 Combine like terms.


So the answer is x=3, y=5, and z=-2 which forms the point (3,5,-2)