SOLUTION: Can you please help me solve this linear equation using substitution -3x+6y = 9 5x-2y = -7 Thank you.

Algebra ->  Coordinate Systems and Linear Equations  -> Lessons -> SOLUTION: Can you please help me solve this linear equation using substitution -3x+6y = 9 5x-2y = -7 Thank you.      Log On


   

Question 702162: Can you please help me solve this linear equation using substitution
-3x+6y = 9
5x-2y = -7
Thank you.
Answer by MathLover1(20850) 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

-3%2Ax%2B6%2Ay=9
5%2Ax-2%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

6%2Ay=9%2B3%2AxAdd 3%2Ax to both sides

y=%289%2B3%2Ax%29%2F6 Divide both sides by 6.


Which breaks down and reduces to



y=3%2F2%2B%281%2F2%29%2Ax Now we've fully isolated y

Since y equals 3%2F2%2B%281%2F2%29%2Ax we can substitute the expression 3%2F2%2B%281%2F2%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


5%2Ax%2B-2%2Ahighlight%28%283%2F2%2B%281%2F2%29%2Ax%29%29=-7 Replace y with 3%2F2%2B%281%2F2%29%2Ax. Since this eliminates y, we can now solve for x.

5%2Ax-2%2A%283%2F2%29-2%281%2F2%29x=-7 Distribute -2 to 3%2F2%2B%281%2F2%29%2Ax

5%2Ax-6%2F2-%282%2F2%29%2Ax=-7 Multiply



5%2Ax-3-1%2Ax=-7 Reduce any fractions

5%2Ax-1%2Ax=-7%2B3Add 3 to both sides


5%2Ax-1%2Ax=-4 Combine the terms on the right side



4%2Ax=-4 Now combine the terms on the left side.


cross%28%281%2F4%29%284%2F1%29%29x=%28-4%2F1%29%281%2F4%29 Multiply both sides by 1%2F4. This will cancel out 4%2F1 and isolate x

So when we multiply -4%2F1 and 1%2F4 (and simplify) we get



x=-1 <---------------------------------One answer

Now that we know that x=-1, lets substitute that in for x to solve for y

5%28-1%29-2%2Ay=-7 Plug in x=-1 into the 2nd equation

-5-2%2Ay=-7 Multiply

-2%2Ay=-7%2B5Add 5 to both sides

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

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

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


y=1 Reduce


So this is the other answer


y=1<---------------------------------Other answer


So our solution is

x=-1 and y=1

which can also look like

(-1,1)

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

-3%2Ax%2B6%2Ay=9
5%2Ax-2%2Ay=-7

we get


graph of -3%2Ax%2B6%2Ay=9 (red) and 5%2Ax-2%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 (-1,1). This verifies our answer.


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

Plug in (-1,1) into the system of equations


Let x=-1 and y=1. Now plug those values into the equation -3%2Ax%2B6%2Ay=9

-3%2A%28-1%29%2B6%2A%281%29=9 Plug in x=-1 and y=1


3%2B6=9 Multiply


9=9 Add


9=9 Reduce. Since this equation is true the solution works.


So the solution (-1,1) satisfies -3%2Ax%2B6%2Ay=9



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

5%2A%28-1%29-2%2A%281%29=-7 Plug in x=-1 and y=1


-5-2=-7 Multiply


-7=-7 Add


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


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


Since the solution (-1,1) satisfies the system of equations


-3%2Ax%2B6%2Ay=9
5%2Ax-2%2Ay=-7


this verifies our answer.