Simultaneous Equations — 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.

Solving a system produces the equation $0 = 0$ at the final step. State how many solutions the system has, what this means geometrically, and what must be given as the answer.

Determine the nature of the solutions of $2x - y + z = 4$, $4x - 2y + 2z = 8$, $x + y - z = 1$, giving the full solution set and a geometric description.

Prove that a system of linear equations cannot have exactly two solutions, and use the argument to explain why the solution set of any consistent linear system must be a point, a line, or a plane.

Two equations of a system are $3x - 6y + 9z = 12$ and $x - 2y + 3z = 7$. What does this tell you about the system's solutions, and why?

For what value of $p$ does the system $x + y + 2z = 3$, $2x - y + z = 1$, $3x + 0y + 3z = p$ have infinitely many solutions? What happens for other values of $p$, and what is the geometry in each case?

Analyse the system $x + y + z = 1$, $2x + 3y + kz = 3$, $x + ky + 3z = 2$ completely: determine all values of $k$ giving a unique solution, infinitely many, or none, with the general solution or the inconsistency in each special case.