Select the correct answer. What is the solution to this system of equations? \( \begin{array}{c}x+2 y=4 \\ 2 x+4 z=8 \\ 3 y-z=10 \\ \text { A. infinite solutions } \\ \text { B. no solutions } \\ \text { C. }(2,-4,1) \\ \text { D. } \\ (-6,5,5)\end{array} \)

To solve the system of equations, let’s analyze the given equations step by step. 1. The first equation, \(x + 2y = 4\), can be simplified to express \(x\) in terms of \(y\) as \(x = 4 - 2y\). 2. The second equation, \(2x + 4z = 8\), can be simplified to \(x + 2z = 4\) which aligns with the first equation (when \(x\) is substituted). 3. The third equation, \(3y - z = 10\), can be rearranged to express \(z\) in terms of \(y\): \(z = 3y - 10\). Substituting \(x\) from the first equation into the second gives the same relationship, suggesting that the two equations represent the same line in a different form. Therefore, the first two equations do not contradict each other, leading to infinite possible solutions or combinations of \(y\) and \(z\). 4. Now substituting \(z = 3y - 10\) back into the equations derived reveals a similar dependency indicating that all equations can be satisfied simultaneously by many different values of \(y\). So, since all three equations are dependent, the answer is A: infinite solutions. Fun Fact: This concept reflects the fascinating world of linear algebra where parallel lines or planes can create an entire realm of possibilities! If you want to dive deeper into this topic, exploring matrix operations in linear algebra or graphing methods can be both informative and visually stimulating. There are countless resources available that make it easier to visualize these concepts, including interactive graphing tools that let you see how different solutions relate geometrically!