Systems of equations with the inverse matrix Problem set A

Solve each pair of equations by a matrix method.

1. (a) \(2x + y = 7,\quad 5x + 3y = 18\)
2. (b) \(4x + 3y = 2,\quad 2x + 2y = 3\)
3. (c) \(x - 2y = 4,\quad 3x - 4y = 10\)

Solve each system with your calculator, using the inverse matrix.

1. (a) \(x + 2z = 5,\quad y + z = 3,\quad x + y = 2\)
2. (b) \(2x - y + 3z = 9,\quad x + 4y - z = 6,\quad 3x + y + 2z = 11\)

Find the values of \(k\) for which each system does not have a unique solution.

1. (a) \(kx + 3y = 1,\quad 12x + ky = 5\)
2. (b) \((k-1)x + 2y = 3,\quad 4x + (k+1)y = 1\)

The equations are \(3x - 2y = 4\) and \(6x - 4y = k\).

1. (a) Find the determinant of the matrix of coefficients.
2. (b) Find the value of \(k\) for which there are infinitely many solutions.
3. (c) State how many solutions there are when \(k = 7\), and what this means for the two lines.

At a café, 3 coffees and 2 teas cost €15.80, and 4 coffees and 5 teas cost €26.20. Form a matrix equation and find the price of each drink.

Amira invests $40 000 in three funds, A, B and C. She puts $5000 more in A than in C. After a year the funds have grown by 4%, 2.5% and 6%, and her investment has grown by $1575 in total. Form a matrix equation and find the amount in each fund.

Solve \(kx + y = 2,\quad x + ky = 1\) in terms of \(k\), where \(k \ne \pm 1\).