Question

Identify the number of solutions of the system of linear equations. 5x+y−z=6 x+y+z=2 12x+4y=10

Answer 1

x =
`(19)/(12)` y =
`(-(9)/(4) )` z =
`(8)/(3)` One solution for each variable.

Explanation:

5x + y − z = 6

x + y + z = 2

12x + 4y = 10

The first thing we need to do is solve for x in the 3rd equation because it inly have 2 variables, x and y.

12x + 4y = 10 Subtract 4y from each side

12x + 4y - 4y = 10 - 4y

12x = 10 - 4y Pull 2 out on the right side

12x = 2(5 - 2y) Divide each side by 12

12x/12 = 2(5 - 2y)/12

x = 2(5 - 2y)/12

x = (5 - 2y)/6

Now we plug in our x value into the 2nd equation and solve for z

x + y + z = 2

`(5-2y)/(6)` + y + z = 2 Multiply each side by 6

6(
`(5-2y)/(6)` + y + z) = 2 * 6

6(
`(5-2y)/(6)` + y + z) = 12

5 - 2y + 6y + 6z = 12 Combine like terms

5 + 4y + 6z = 12 Subtract 5 from each side

5 - 5 + 4y + 6z = 12 - 5

4y + 6z = 7 Subtract 4y from each side

4y - 4y + 6z = 7 - 4y

6z = 7 - 4y Divide each side by 6

6z/6 = (7 - 4y)/6

z = (7 - 4y)/6

Now we solved for z and x, so in the 1st equation we plug in x and z.

5x + y − z = 6

5(
`(5-2y)/(6)`) + y -
`(7-4y)/(6)` = 6 Multiply each side by 6

6*(5(
`(5-2y)/(6)`) ) + 6y - 6(
`(7-4y)/(6)`) = 6*6

6*(5(
`(5-2y)/(6)`) ) + 6y - 6(
`(7-4y)/(6)`) = 36

5(5 - 2y) + 6y - 7 - 4y = 36

25 - 10y + 6y - 7 - 4y = 36 Rearrange to make it easier to combine terms.

25 - 7 - 10y + 6y - 4y = 36

18 - 8y = 36 Subtract 18 from each side.

18 - 18 - 8y = 36 - 18

- 8y = 36 - 18

- 8y = 18 Divide each side by -8

- 8y/-8 = 18/- 8

y = 18/- 8

y = - 9/4

Now we plug our answer for y back into the 3rd equation and solve for the value of x.

12x + 4y = 10

12x + 4
`(-(9)/(4) )` = 10

12x - 9 = 10 Add 9 to each side

12x - 9 + 9 = 10 + 9

12x = 10 + 9

12x = 19 Divide each side by 12

12x/12 = 19/12

x = 19/12

Now we have a value for x and y so plug these into the 2nd equation to sovle for z.

x + y + z = 2

`(19)/(12)` +
`(-(9)/(4) )` + z = 2 We need to find the common denominator in order to add.

`(-(9)/(4) )` *
`(3)/(3)` =
`-(27)/(12)`

`(19)/(12)`
`-(27)/(12)` + z = 2

`-(8)/(12)` + z = 2 Add
`-(8)/(12)` to each side

`-(8)/(12)`
`+ (8)/(12)` + z = 2
`+ (8)/(12)`

z = 2
`+ (8)/(12)` Reduce
`+ (8)/(12)` to
`(2)/(3)`

z = 2 +
`(2)/(3)` To add find a common denominator.

`2 * (3)/(3) = (6)/(3)`

z =
`(6)/(3)` +
`(2)/(3)`

z =
`(8)/(3)`

So there is 1 solution for each variable.