Solve the given linear system a y 5x 3 y 3x 1 b 2x 3y

Solve the given linear system.

a. y = 5x - 3
y = 3x - 1

b. -2x + 3y = 8
3x + 5y = -12

c. 2x - 4y = -6
-x + 2y = 3

d. 5x + 10y = 70
5x + 25z = 270
10y + 25z = 300

e. 2x - y - z = 5
x + y + z = 7
3x - 2y - 3z = 1

Solution

a. Use substitution to solve (substitute for y and set them equal to each other)
y = 5x - 3
y = 3x - 1

5x - 3 = 3x - 1
-3x -3x
______________
2x - 3 = - 1
+3 +3
______________
2x = 2
x = 1
Plug in the value of x to find y
y = 5(1) - 3
y = 2

(1,2)
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b. Use elimination to solve the problem (Try to get rid of one variable I picked x to get rid of)
(multiplying by 3 to the top Eq. & multiplying by 2 to the bottom Eq.)
3 ( -2x + 3y = 8 )
2 ( 3x + 5y = -12 )
_________________

-6x + 9y = 24
6x +10y = -24
_________________ now x can be eliminated.
19y = 0
y = 0
Plug in the value of y to find x
-2x + 3y = 8
-2x + 3(0) = 8
-2x = 8
x = -4

(-4,0)
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c. For this problem i will use substitution sove for x for the 2nd equation
2x - 4y = -6
-x + 2y = 3

-x + 2y = 3
+x +x
______________
2y = 3 + x
-3 -3
______________
x = 2y - 3 Now Plug in 2y -3 in the x for the 1st equation

2x - 4y = -6
2 (2y - 3) - 4y = -6
4y - 6 -4y = -6
+6 +6
0y = 0
0 = 0
when you have a 0 = 0 that means infinite many solution for this 2 equations ( coincided lines.)

infinite many solution

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d. I took 1st 2 equation and try to eliminate a variable x by multiplying by (-1) to 1st equation
5x + 10y = 70
5x + 25z = 270
10y + 25z = 300

-1 (5x + 10y = 70) >>>>>> -5x - 1oy = -70
5x + 25z = 270 >>>>>>>>> 5x + 25z = 270

-5x - 1oy = -70
5x + 25z = 270

_______________ now x is eliminated and new equation will be form
-10y + 25z = 200 ( now take the 3rd equation and do elimination y.
10y + 25z = 300
____________________
50z = 500

z = 10 plug in z to find y

10y + 25z = 300
10y + 25(10) = 300
10y + 250 = 300
-250 -250
10y = 50

y = 5 plug in y to find x

5x + 10(5) = 70

5x + 50 = 70
-50 -50
5x = 20
x = 4

(4, 5, 10)

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e. Eliminate z & y from first 2 equation
2x - y - z = 5
x + y + z = 7
3x - 2y - 3z = 1


2x - y - z = 5
x + y + z = 7
__________

3x = 12
x = 4 Now take 2nd and 3rd equation and plug in x

x + y + z = 7 >>>>>>>>>>> 4 + y + z = 7 >>>>>>> y + z = 3
3x - 2y - 3z = 1 >>>>>>>>>> 3(4) - 2y - 3z = 1 >>>>> - 2y - 3z = -11


y + z = 3
- 2y - 3z = -11 Now eliminate y by mulitplying by 2 to the 1st equation

2( y + z = 3 )
- 2y - 3z = -11

2y + 2z = 6
- 2y - 3z = -11
_____________
-z = -5

z = 5 Now plug in z and x value to find y

x + y + z = 7
4 + y + 5 = 7
9 + y = 7
-9 -9
y = -2

(4, -2, 5)