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Chapter 5: Problem 18

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

Short Answer

Expert verified
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 by step solution

01

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 \( z = \frac{2(x-3y)+3}{7} \)
02

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.
03

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 \( z \).
04

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.

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Most popular questions from this chapter

will help you prepare for the material covered in the next section. Graph \(x-y=3\) and \((x-2)^{2}+(y+3)^{2}=4\) in the same rectangular coordinate system. What are the two intersection points? Show that each of these ordered pairs satisfies both equations.

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