Systems of 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.

Draw the region satisfying $3x + 4y < 24$ with $x \geq 0$ and $y \geq 0$. State the intercepts, the line style, and how you decided which side to shade.

A shaded region on a graph has a solid boundary line passing through $(0, 5)$ and $(10, 0)$, and the shaded area lies above the line. Write the inequation, and explain how each feature of the graph determined a feature of your answer.

Explain why the inequality sign reverses when both sides are multiplied by a negative number, and use your reasoning to explain why testing a point is more reliable than rearranging an inequation into the form $y \leq \dots$ before graphing.

Write the system of inequations for this situation: a delivery van carries boxes weighing 20 kg and crates weighing 35 kg. The van can carry at most 700 kg and at most 25 items in total. At least 5 crates must be carried.

A region is defined by $x + y \leq 10$, $2x + y \leq 16$, $x \geq 0$, $y \geq 2$. Find all corner points of the feasible region, verifying each, and describe what each corner represents if $x$ and $y$ are hours spent on two tasks.

A caterer must prepare at least 60 servings using two dishes. Dish A serves 4 people and takes 20 minutes; dish B serves 6 people and takes 45 minutes. Only 6 hours of preparation time is available, and at most 12 dishes can fit in the kitchen. Determine whether the caterer's requirements can all be met, and analyse how the answer changes if the serving requirement rises.

A farmer buys 15 bags of feed, some at $28 each and the rest at $45 each, spending $536 in total. Form a system of equations and find how many of each kind were bought.

A courtyard is to be paved with a rectangular area of paving surrounded by a border of uniform width. The paved rectangle measures 12 m by 8 m, and the total area including the border is 165 m². Find the width of the border, and explain why only one of your solutions is used.

A fruit grower sells apples at $4/kg and pears at $6/kg. Last week the total revenue was $1,200 from 250 kg of fruit. This week the grower wants revenue of at least $1,400 from at most 260 kg, and can pick at most 180 kg of apples. Determine whether this week's targets are achievable, and analyse the trade-off the grower faces.