Simultaneous Equations — practice questions

Solve the system $2x + 5y = 24$ and $x - 3y = -10$, and check your answer.

A café sells flat whites and muffins. Three flat whites and two muffins cost $26.50; five flat whites and four muffins cost $47.50. Find the price of each, and explain how the graph of the system relates to your answer.

For what value(s) of $k$ does the system $3x + ky = 12$ and $6x + 8y = 24$ have (i) exactly one solution, (ii) no solution, (iii) infinitely many solutions? Justify each case both algebraically and graphically.

Solve the system $y = x^2 - 3x + 4$ and $y = x + 1$, giving your answers as coordinate pairs.

A rectangular vegetable plot is to be fenced against an existing wall, so fencing is needed on only three sides. There is 24 m of fencing available and the plot must have an area of 70 m². Find the possible dimensions, and explain what the two answers mean for the gardener.

The line $y = mx + 4$ and the curve $y = x^2 + 3x + 8$ are given. Determine all values of $m$ for which the line and the curve (i) meet at two points, (ii) touch at exactly one point, (iii) never meet. Interpret the boundary cases geometrically.