The system of equations in the image is:

$1) \quad x + 5y + 7z = 0$ $2) \quad x + y + 7z = 4$ $3) \quad -3y + 7z = 3$

Solution:

We can solve this system step by step. Let’s start with equation (3):

$-3y + 7z = 3$

Solve for $y$:

$y = \frac{7z - 3}{3}$

Now substitute this expression for $y$ into the other two equations.

Substituting into Equation (2):

Substitute $y = \frac{7z - 3}{3}$ into the second equation:

$x + \left(\frac{7z - 3}{3}\right) + 7z = 4$

Multiply through by 3 to eliminate the fraction:

$3x + (7z - 3) + 21z = 12$

Simplify:

$3x + 28z - 3 = 12$ $3x + 28z = 15$ $x = \frac{15 - 28z}{3}$

Substituting into Equation (1):

Now substitute $y = \frac{7z - 3}{3}$ into the first equation:

$x + 5\left(\frac{7z - 3}{3}\right) + 7z = 0$

Multiply through by 3 to eliminate the fraction:

$3x + 5(7z - 3) + 21z = 0$ $3x + 35z - 15 + 21z = 0$ $3x + 56z = 15$ $x = \frac{15 - 56z}{3}$

Now we have two expressions for $x$:

$x = \frac{15 - 28z}{3}$ $x = \frac{15 - 56z}{3}$

Since both equal $x$, we can set them equal to each other:

$\frac{15 - 28z}{3} = \frac{15 - 56z}{3}$

Multiplying both sides by 3:

$15 - 28z = 15 - 56z$

Cancel out the 15 on both sides:

$-28z = -56z$

Simplify:

$28z = 56z$

Solve for $z$:

$z = 0$

Now substitute $z = 0$ back into the expressions for $y$ and $x$:

From $y = \frac{7z - 3}{3}$:

$y = \frac{7(0) - 3}{3} = -1$

From $x = \frac{15 - 28z}{3}$:

$x = \frac{15 - 28(0)}{3} = 5$

Final Solution:

$x = 5, \quad y = -1, \quad z = 0$

Would you like more details or further explanation?

Related Questions:

1. How can systems of linear equations be solved using matrices?
2. What is the method of substitution in solving systems of equations?
3. Can we solve this system using Gaussian elimination?
4. How does the determinant of the coefficient matrix affect the solution of a system of equations?
5. What is Cramer's rule and how does it apply to this system?

Tip:

Always check your solution by substituting the values of the variables back into the original equations to verify correctness.