Answer:
x = -8, y = -4, z = 9
Step-by-step explanation:
Certainly! To solve the system of linear equations, we need to find the values of x, y, and z that satisfy all three equations.
The given equations are:
1) -3x + 3y = 12
2) -3x + 4y + z = 17
3) 3x + 3y + 4z = 19
To solve this system of equations, we can proceed as follows:
From the first equation:
-3x + 3y = 12
x = (3y - 12) / -3
Next, we will substitute the value of x into the second and third equations:
In the second equation:
-3((3y - 12) / -3) + 4y + z = 17
3(y - 4) + 4y + z = 17
3y - 12 + 4y + z = 17
7y - 12 + z = 17
7y + z = 29
In the third equation:
3x + 3y + 4z = 19
3((3y - 12) / -3) + 3y + 4z = 19
3(y - 4) + 3y + 4z = 19
3y - 12 + 3y + 4z = 19
6y - 12 + 4z = 19
6y + 4z = 31
Now we have two equations:
1) 7y + z = 29
2) 6y + 4z = 31
From here, we can solve for y and z. After finding y and z, we can substitute them back into the equation x = (3y - 12) / -3 to find the value of x. Finally, we get:
x = -8, y = -4, z = 9
So, the solution to the system of linear equations is:
x = -8, y = -4, z = 9
Answer:
[tex]\sf x = \dfrac{3}{2}[/tex]
[tex]\sf y = \dfrac{11}{2}[/tex]
[tex]\sf z = -\dfrac{1}{2}[/tex]
Step-by-step explanation:
Let's solve the system of equations using the substitution method:
[tex]\sf \begin{cases} -3x + 3y = 12 \\ -3x + 4y + z = 17 \\ 3x + 3y + 4z = 19 \end{cases} [/tex]
From the first equation, we can express [tex]\sf x[/tex] in terms of [tex]\sf y[/tex]:
[tex]\sf -3x + 3y = 12 [/tex]
[tex]\sf -3x = -3y + 12 [/tex]
[tex]\sf x = y - 4 [/tex]
Now, substitute [tex]\sf x[/tex] into the second equation:
[tex]\sf -3(y - 4) + 4y + z = 17 [/tex]
[tex]\sf -3y + 12 + 4y + z = 17 [/tex]
[tex]\sf y + z = 5 [/tex]
Now, substitute [tex]\sf y[/tex] from the above result into the third equation:
[tex]\sf 3(y - 4) + 3y + 4z = 19 [/tex]
[tex]\sf 3y - 12 + 3y + 4z = 19 [/tex]
[tex]\sf 6y + 4z = 31 [/tex]
Now, we have two equations:
1. [tex]\sf y + z = 5[/tex]
2. [tex]\sf 6y + 4z = 31[/tex]
From equation (1), we can express [tex]\sf y[/tex] in terms of [tex]\sf z[/tex]:
[tex]\sf y = 5 - z [/tex]
Now, substitute [tex]\sf y[/tex] into equation (2):
[tex]\sf 6(5 - z) + 4z = 31 [/tex]
[tex]\sf 30 - 6z + 4z = 31 [/tex]
[tex]\sf -2z = 1 [/tex]
[tex]\sf z = -\dfrac{1}{2} [/tex]
Now that we have [tex]\sf z = -\dfrac{1}{2}[/tex], substitute [tex]\sf z[/tex] into equation (1) to find [tex]\sf y[/tex]:
[tex]\sf y - \dfrac{1}{2} = 5 [/tex]
[tex]\sf y = \dfrac{11}{2} [/tex]
Finally, substitute [tex]\sf y[/tex] and [tex]\sf z[/tex] back into the expression for [tex]\sf x[/tex] we found earlier:
[tex]\sf x = \dfrac{11}{2} - 4 [/tex]
[tex]\sf x = \dfrac{3}{2} [/tex]
So, the solution is [tex]\sf x = \dfrac{3}{2}[/tex], [tex]\sf y = \dfrac{11}{2}[/tex], and [tex]\sf z = -\dfrac{1}{2}[/tex].
