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

Answer:

x = $\frac{19}{12}$     y = $(-\frac{9}{4} )$   z = $\frac{8}{3}$    One solution for each variable.

Step-by-step 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

$\frac{5-2y}{6}$ + y + z = 2    Multiply each side by 6

6($\frac{5-2y}{6}$ + y + z) = 2 * 6

6($\frac{5-2y}{6}$ + 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($\frac{5-2y}{6}$) + y - $\frac{7-4y}{6}$ = 6     Multiply each side by 6

6*(5($\frac{5-2y}{6}$) ) + 6y - 6($\frac{7-4y}{6}$) = 6*6

6*(5($\frac{5-2y}{6}$) ) + 6y - 6($\frac{7-4y}{6}$) = 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$(-\frac{9}{4} )$ = 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

$\frac{19}{12}$ + $(-\frac{9}{4} )$ + z = 2      We need to find the common denominator in order to add.

$(-\frac{9}{4} )$ * $\frac{3}{3}$ = $-\frac{27}{12}$

$\frac{19}{12}$ $-\frac{27}{12}$ + z = 2

$-\frac{8}{12}$ + z = 2   Add $-\frac{8}{12}$ to each side

$-\frac{8}{12}$  $+ \frac{8}{12}$ + z = 2 $+ \frac{8}{12}$

z =  2 $+ \frac{8}{12}$       Reduce   $+ \frac{8}{12}$  to $\frac{2}{3}$

z = 2 + $\frac{2}{3}$    To add find a common denominator.

$2 * \frac{3}{3} = \frac{6}{3}$

z = $\frac{6}{3}$ + $\frac{2}{3}$

z = $\frac{8}{3}$

So there is 1 solution for each variable.