SOLUTION: Use substitution to solve the system. (Simplify your answer completely. If the system is dependent, enter a general solution in terms of x and y. If there is no solution, enter NO

Algebra ->  Coordinate Systems and Linear Equations  -> Lessons -> SOLUTION: Use substitution to solve the system. (Simplify your answer completely. If the system is dependent, enter a general solution in terms of x and y. If there is no solution, enter NO       Log On


   

Question 1130674: Use substitution to solve the system. (Simplify your answer completely. If the system is dependent, enter a general solution in terms of x and y. If there is no solution, enter NO SOLUTION.)
5x-2 / 4 + 1/2 = 3y+2 / 2
7y+3 / 3 = x/2 + 7/3
Found 2 solutions by MathLover1, MathTherapy:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

%285x-2+%29%2F+4+%2B+1%2F2+=+%283y%2B2%29+%2F+2+....both side multiply by 4
%287y%2B3%29+%2F+3+=+x%2F2+%2B+7%2F3...both side multiply by 3

4%285x-2+%29%2F+4+%2B+1%2A4%2F2+=+4%283y%2B2%29+%2F+2+
6%287y%2B3%29+%2F+3+=+6x%2F2+%2B+%286%2A7%29%2F3
--------------------------------------------
5x-2++%2B+2+=+2%283y%2B2%29++
2%287y%2B3%29++=+3x+%2B+2%2A7
---------------------------------
5x=+6y%2B4++
14y%2B6++=+3x+%2B+14
------------------------------
5x-6y=4++
-3x%2B14y=++8


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


Lets start with the given system of linear equations

5%2Ax-6%2Ay=4
-3%2Ax%2B14%2Ay=8

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=4-5%2AxSubtract 5%2Ax from both sides

y=%284-5%2Ax%29%2F-6 Divide both sides by -6.


Which breaks down and reduces to



y=-2%2F3%2B%285%2F6%29%2Ax Now we've fully isolated y

Since y equals -2%2F3%2B%285%2F6%29%2Ax we can substitute the expression -2%2F3%2B%285%2F6%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


-3%2Ax%2B14%2Ahighlight%28%28-2%2F3%2B%285%2F6%29%2Ax%29%29=8 Replace y with -2%2F3%2B%285%2F6%29%2Ax. Since this eliminates y, we can now solve for x.

-3%2Ax%2B14%2A%28-2%2F3%29%2B14%285%2F6%29x=8 Distribute 14 to -2%2F3%2B%285%2F6%29%2Ax

-3%2Ax-28%2F3%2B%2870%2F6%29%2Ax=8 Multiply



-3%2Ax-28%2F3%2B%2835%2F3%29%2Ax=8 Reduce any fractions

-3%2Ax%2B%2835%2F3%29%2Ax=8%2B28%2F3Add 28%2F3 to both sides


-3%2Ax%2B%2835%2F3%29%2Ax=24%2F3%2B28%2F3 Make 8 into a fraction with a denominator of 3


-3%2Ax%2B%2835%2F3%29%2Ax=52%2F3 Combine the terms on the right side



%28-9%2F3%29%2Ax%2B%2835%2F3%29x=52%2F3 Make -3 into a fraction with a denominator of 3

%2826%2F3%29%2Ax=52%2F3 Now combine the terms on the left side.


cross%28%283%2F26%29%2826%2F3%29%29x=%2852%2F3%29%283%2F26%29 Multiply both sides by 3%2F26. This will cancel out 26%2F3 and isolate x

So when we multiply 52%2F3 and 3%2F26 (and simplify) we get



x=2 <---------------------------------One answer

Now that we know that x=2, lets substitute that in for x to solve for y

-3%282%29%2B14%2Ay=8 Plug in x=2 into the 2nd equation

-6%2B14%2Ay=8 Multiply

14%2Ay=8%2B6Add 6 to both sides

14%2Ay=14 Combine the terms on the right side

cross%28%281%2F14%29%2814%29%29%2Ay=%2814%2F1%29%281%2F14%29 Multiply both sides by 1%2F14. This will cancel out 14 on the left side.

y=14%2F14 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=2 and y=1

which can also look like

(2,1)

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

5%2Ax-6%2Ay=4
-3%2Ax%2B14%2Ay=8

we get


graph of 5%2Ax-6%2Ay=4 (red) and -3%2Ax%2B14%2Ay=8 (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 (2,1). This verifies our answer.


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

Plug in (2,1) into the system of equations


Let x=2 and y=1. Now plug those values into the equation 5%2Ax-6%2Ay=4

5%2A%282%29-6%2A%281%29=4 Plug in x=2 and y=1


10-6=4 Multiply


4=4 Add


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


So the solution (2,1) satisfies 5%2Ax-6%2Ay=4



Let x=2 and y=1. Now plug those values into the equation -3%2Ax%2B14%2Ay=8

-3%2A%282%29%2B14%2A%281%29=8 Plug in x=2 and y=1


-6%2B14=8 Multiply


8=8 Add


8=8 Reduce. Since this equation is true the solution works.


So the solution (2,1) satisfies -3%2Ax%2B14%2Ay=8


Since the solution (2,1) satisfies the system of equations


5%2Ax-6%2Ay=4
-3%2Ax%2B14%2Ay=8


this verifies our answer.

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

Use substitution to solve the system. (Simplify your answer completely. If the system is dependent, enter a general solution in terms of x and y. If there is no solution, enter NO SOLUTION.)
5x-2 / 4 + 1/2 = 3y+2 / 2
7y+3 / 3 = x/2 + 7/3
If this is: , then follow on. 

5x - 2 + 2 = 2(3y + 2) -------- Multiplying eq (i) by LCD, 4

5x = 6y + 4

5x - 6y = 4 ------ eq (iii)



2(7y + 3) = 3x + 14 -------- Multiplying eq (ii) by LCD, 6

14y + 6 = 3x + 14 

3x - 14y = - 8 --- eq (iv)

2x + 8y = 12 ------ Subtracting eq (iv) from eq (iii)

2(x + 4y) = 2(6)

x + 4y = 6______x = 6 - 4y ------ eq (v)

5(6 - 4y) - 6y = 4 ------- Substituting 6 - 4y for x in eq (iii)

30 - 20y - 6y = 4

- 26y = - 26

 



5x - 6(1) = 4 ------- Substituting 1 for y in eq (iii)

5x - 6 = 4

5x = 10

highlight_green%28matrix%281%2C5%2C+x%2C+%22=%22%2C+10%2F5%2C+%22=%22%2C+2%29%29