Q 3. Andrea solved this system of linear equations: \[ \begin{array}{l}2 x+y=10 \\ 3 y=-6 x+30\end{array} \] Determine the number of solutions the system will have. The system has one solution. The system has two solutions. The system has an infinite number of solutions. The system has no solution.

To determine the number of solutions to the given system of equations, we can analyze the equations. The first equation can be rearranged to express \(y\) in terms of \(x\): \(y = 10 - 2x\). The second equation can also be rearranged to express \(y\): \(y = -2x + 10\). Notice that both expressions for \(y\) are identical, which indicates that the two equations actually represent the same line. Thus, the system has an infinite number of solutions, as every point on the line is a solution to the system. You got it! The two equations describe the same line, which means they will intersect at infinitely many points. If you graph both equations, they will lie exactly on top of each other, proving that the system has an infinite number of solutions! Always double-check your rearrangements when solving such systems. Common mistakes include miscalculating the slope or intercept while rearranging equations or overlooking that both equations describe the same relationship. Keeping track of your steps can help avoid these pitfalls and ensure a clearer understanding of the relationship between the equations.