Solving Systems
Is (2, 3) a solution of both equations: x + y = 5 and x - y = -1?
Yes. It satisfies both equations.
Is (2, 1) in the feasible region defined by x + y ≤ 6, x ≥ 0, and y ≥ 0?
Yes. All three inequalities are satisfied.
A matrix has 2 rows and 3 columns. State its dimensions.
2 × 3
True or false: A solution of a system must satisfy every equation in that system at the same time.
True.
Solve the system: y = x + 1; 2x + y = 10.
(x, y) = (3, 4)
A booth sells x caps and y scarves. It can sell at most 12 items and has only 5 caps. Write the constraints, including nonnegativity.
x + y ≤ 12; x ≤ 5; x ≥ 0; y ≥ 0. Both variables must be whole numbers.
The rows of matrix A are [2, -1, 4, 6], [0, 3, 5, 8], and [-2, 7, 1, 9]. Find a₂₃ and a₃₁.
a₂₃ = 5; a₃₁ = -2
Can (2, 4) be a solution of the system x < 2 and y ≥ 1? Explain briefly.
No. Its x-coordinate equals 2, but x must be less than 2. The dashed boundary is excluded.
Classify the system as having one, no, or infinitely many solutions: x + 2y = 7; 3x + 6y = 18.
No solution. Multiplying the first equation by 3 gives 3x + 6y = 21, not 18.
List all vertices of the region: x + y ≤ 12; x ≤ 7; x ≥ 0; y ≥ 0.
(0, 0), (7, 0), (7, 5), and (0, 12)
A matrix has rows R₁ = [1, 2, 5] and R₂ = [3, 4, 11]. Apply R₂ → R₂ - 3R₁. State the new second row.
R₂ = [0, -2, -4]
An augmented matrix reduces to rows [1, 0 | 4] and [0, 0 | 0]. How many solutions are there, and what can you say about x and y?
Infinitely many solutions. x = 4 and y can be any real number.
Solve for (x, y, z): x + y + z = 6; 2x - y + z = 7; x + 2y - z = 3.
(x, y, z) = (3, 1, 2)
A feasible region has vertices (0, 0), (6, 0), (6, 2), and (0, 8). Find the maximum value of P = 4x + 7y and where it occurs.
Maximum P = 56 at (0, 8). The four vertex values are 0, 24, 38, and 56.
A matrix has rows [1, 2, -1, 4], [0, 1, 3, 2], and [2, -1, 1, 5]. First apply R₃ → R₃ - 2R₁, then R₂ → R₂ + R₁. State all three resulting rows.
R₁ = [1, 2, -1, 4]; R₂ = [1, 3, 2, 6]; R₃ = [0, -5, 3, -3].
A student multiplies 2x + y = 5 by 3 and writes 6x + y = 15. Is the new equation equivalent? Correct the error.
No. Every term must be multiplied by 3. The equivalent equation is 6x + 3y = 15.
Solve for (x, y, z): x + 2y + z = 11; 2x - y + z = 8; 3x + y - z = 7.
(x, y, z) = (3, 2, 4)
A baker sells x muffins and y mini loaves. At most 16 items can be sold, with no more than 10 muffins. Profit is $5 per muffin and $3 per loaf. Write the constraints, list the feasible vertices, and find the maximum profit. Quantities are nonnegative whole numbers.
Constraints: x + y ≤ 16; x ≤ 10; x ≥ 0; y ≥ 0. Vertices: (0, 0), (10, 0), (10, 6), (0, 16). Maximum profit: $68 at (10, 6), selling 10 muffins and 6 mini loaves.
Use an augmented matrix to solve x + 2y = 9 and 3x - y = 13. Give the RREF and the ordered-pair solution.
RREF rows: [1, 0 | 5] and [0, 1 | 2]. Solution: (x, y) = (5, 2).
A shipment has 60 boxes in three sizes. Small boxes weigh 2 kg, medium boxes 3 kg, and large boxes 5 kg. The boxes weigh 195 kg in total, and there are twice as many medium boxes as small boxes. Let x, y, z represent the small, medium, and large box counts. Write a system of three equations. Do not solve.
x + y + z = 60; 2x + 3y + 5z = 195; y = 2x.