Problem 18 - College Algebra Essentials

Question

Solve each system. $$\left\\{\begin{aligned} 7 z-3 &=2(x-3 y) \\ 5 y+3 z-7 &=4 x \\ 4+5 z &=3(2 x-y) \end{aligned}\right.$$

Step-by-Step Solution

Answer

To obtain the final solution, perform each of the steps iteratively. Firstly, solve for \( z \) in the first equation then substitute this value into the other equations. Solve that system to get \( x \) and \( y \). Lastly, substitute \( x \) and \( y \) back into any of the original equations to attain \( z \). The obtained \( x \), \( y \), \( z \) values are the solution for the system.

Step 1: Isolate One Variable

Firstly, isolate one variable in the first equation to simplify the procedure. In this case, isolate \( z \) in the first equation in terms of \( x \) and \( y \). This is done by adding 3 to both sides of the equation and then dividing by 7. This gives \( z = \frac{2(x-3y)+3}{7} \)

Step 2: Substitute Isolated Variable into Other Equations

Substitute the isolated variable \( z \) into the second and third equations. This will create a system of two equations only in terms of variables \( x \) and \( y \), which can then be solved with relative ease.

Step 3: Solve New System of Equations

Now, we have a system involving just two variables, \( x \) and \( y \). Solve this by either substitution or elimination method to find \( x \) and \( y \). After finding \( x \) and \( y \), substitute these values into any of the original equations to find \( z \).

Step 4: Final step: Check Solution

Finally, plug the solution for \( x \), \( y \) and \( z \) back into the original equations to ensure that they are indeed the correct solution to the system.