Solving Systems — practice questions

Solve the system $x + y + z = 6$, $2x - y + z = 3$, $x + 2y - z = 2$, checking your answer in all three equations.

A theatre sells adult, student and child tickets. On opening night it sold 200 tickets for $3,150. Adult tickets are $25, student $15 and child $10. There were three times as many adults as children. Find how many of each were sold, and comment on whether the answer is reasonable.

A quadratic function $y = px^2 + qx + r$ passes through $(1, 6)$, $(3, 20)$ and $(-2, 15)$. Find the function, and explain in general why three points determine a unique quadratic while two do not and four generally cannot be fitted at all.

Write the system $3x - z = 7$, $2x + 4y + z = 1$, $x + y = 5$ as an augmented matrix, and reduce it to row echelon form.

Reduce $\left[\begin{array}{ccc|c} 1 & 2 & 1 & 8 \\ 2 & 3 & -1 & 7 \\ 3 & 5 & 0 & 15 \end{array}\right]$ to row echelon form, state the nature of the solution, and give the general solution if there is more than one.

Explain why every row operation leaves the solution set of a system unchanged, and explain what goes wrong if a row is multiplied by zero or if two columns are swapped.