Solve the following system of equations. 3x+2y-z=2 4x+3y+z=4 7x+5y=6

1. Solve the third equation for $y$<br /> From $7x + 5y = 6$, solve for $y$: $y = \frac{6 - 7x}{5}$.<br />2. Substitute $y$ into the first equation<br /> Substitute $y = \frac{6 - 7x}{5}$ into $3x + 2y - z = 2$: $3x + 2\left(\frac{6 - 7x}{5}\right) - z = 2$.<br />3. Simplify and solve for $z$<br /> Simplify: $15x + 12 - 14x - 5z = 10$. Thus, $x - 5z = -2$. Solve for $z$: $z = \frac{x + 2}{5}$.<br />4. Substitute $y$ and $z$ into the second equation<br /> Substitute $y = \frac{6 - 7x}{5}$ and $z = \frac{x + 2}{5}$ into $4x + 3y + z = 4$: $4x + 3\left(\frac{6 - 7x}{5}\right) + \frac{x + 2}{5} = 4$.<br />5. Simplify and solve for $x$<br /> Simplify: $20x + 18 - 21x + x + 2 = 20$. Thus, $0x + 20 = 20$. This is always true, so $x$ can be any real number.<br />6. Express $y$ and $z$ in terms of $x$<br /> Since $x$ is arbitrary, express $y$ and $z$ as functions of $x$: $y = \frac{6 - 7x}{5}$ and $z = \frac{x + 2}{5}$.