Solving systems of equations Problem set A

Solve without a calculator. \(4x + 3y=-1\) and \(5x - 2y=16\).

Write each equation in the form \(ax^2+bx+c=0\).

1. (a) \((3x+2)(2x-5)=8\)
2. (b) \(2x+\frac{9}{x}=7\)

Solve each system with your calculator. (Draw a graph for part (a) if you wish: the solution is where the lines meet.)

1. (a) \(6x + 5y=-3\), \(2x - 3y=13\)
2. (b) \(y=2x-7\), \(3x+4y=-5\)

Solve using your calculator. \(2x + y - 3z=-6\), \(x - 2y + z=7\), \(3x + y + 2z=11\).

Rearrange into standard form, then solve using your calculator. \(3x + z=2y + 4\), \(x + 4y=2z - 3\), \(y + 3z + 3=-2x + 8\).

Solve using your calculator. \(w + 2x - y + 3z=0\), \(2w - x + y + z=9\), \(3w + x + 2y - 2z=3\), \(w - 3x + y + 2z=14\).

Rearrange into standard form, then solve using your calculator. \(3w + x + z=2y + 5\), \(w - 2x + y + 3z=-4\), \(2x + y=-2w + z + 7\), \(w - 3y + 2z + 2=-x + 5\).

Decide how many solutions each system has: none, one or infinitely many. Give a reason, then answer part (c).

1. (a) \(3x - y = 4\) and \(6x - 2y = 9\)
2. (b) \(2x + 5y = 10\) and \(4x + 10y = 20\)
3. (c) The system \(2x + ky = 5\) and \(6x + 9y = 4\) has no solution. Find \(k\).

A museum charges one price for an adult ticket and another for a child ticket. One group of 2 adults and 5 children pays $61.25. A second group of 3 adults and 4 children pays $66.50.

1. (a) Write two equations.
2. (b) Find the cost of a family ticket for 2 adults and 3 children.

A runner's weekly distances form an arithmetic sequence. The term \(u_{5}\) is 19 and the sum of the first 10 terms is 205. Find the first term \(u_1\) and the common difference \(d\).

A pattern of tiles is built row by row. The sum of the first 4 terms is 14.5 and the sum of the first 9 terms is 49.5. Find the first term \(u_1\) and the common difference \(d\).