Solving systems of equations Problem set A Worked solutions

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

• Label the equations (1) and (2).
• Multiply (1) by 2: \(8x + 6y=-2\)
• Multiply (2) by 3: \(15x - 6y=48\)
• Add the equations to remove y: \(23x=46\)
• So \(x=2\). Put this in (1): \(y=-3\)

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

1. (a) \((3x+2)(2x-5)=8\)

   • Expand the brackets: \(6x^2 - 11x - 10=8\)
   • Subtract 8 from both sides: \(6x^2 - 11x - 18=0\)

2. (b) \(2x+\frac{9}{x}=7\)

   • Multiply every term by \(x\): \(2x^2+9=7x\)
   • Move \(7x\) to the left: \(2x^2 - 7x + 9=0\)

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

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

   • Simultaneous equation solver: x = 2, y = −3
   • Graph check: the two lines meet at (2, −3)

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

   • Standard form: \(2x - y=7\), \(3x + 4y=-5\)
   • Simultaneous equation solver: x = 2.09, y = −2.82

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

• Simultaneous equation solver: x = 2, y = −1, z = 3

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

• Standard form (variables on the left, number on the right): \(3x - 2y + z=4\), \(x + 4y - 2z=-3\), \(2x + y + 3z=5\)
• Simultaneous equation solver: x = 0.714, y = −0.286, z = 1.29
• Exact values: \(x=\frac{5}{7},\ y=-\frac{2}{7},\ z=\frac{9}{7}\)

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

• Simultaneous equation solver: w = 1, x = −2, y = 3, z = 2

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

• Standard form (variables on the left, number on the right): \(3w + x - 2y + z=5\), \(w - 2x + y + 3z=-4\), \(2w + 2x + y - z=7\), \(w + x - 3y + 2z=3\)
• Simultaneous equation solver: w = 0.926, x = 2.50, y = 0.130, z = −0.0185
• Exact values: \(w=\frac{25}{27},\ x=\frac{5}{2},\ y=\frac{7}{54},\ z=-\frac{1}{54}\)

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 \(6x - 2y = 9\)

   • Double the first equation: \(6x - 2y = 8\)
   • The second equation says \(6x - 2y = 9\). The lines are parallel: no solution.

2. (b) \(2x + 5y = 10\) and \(4x + 10y = 20\)

   • Double the first equation: \(4x + 10y = 20\)
   • This is the second equation. The lines are the same: infinitely many solutions.

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

   • Multiply the first equation by 3: \(6x + 3ky = 15\)
   • The left sides must match: \(3k = 9\), so \(k = 3\)
   • The constants differ (15 and 4), so there is no solution, as required.

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.

   • Let \(a\) = price of an adult ticket, \(b\) = price of a child ticket ($).
   • \(2a + 5b=61.25\)
   • \(3a + 4b=66.5\)

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

   • Simultaneous equation solver: a = 12.50, b = 7.25
   • 2a + 3b = 2 × 12.50 + 3 × 7.25 = $46.75

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 \(d\).

• \(u_{5}=u_1+4d=19\)
• \(S_{10}=\frac{10}{2}(2u_1+9d)=205\), so \(10u_1+45d=205\)
• Simultaneous equation solver: u1 = 7, d = 3

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 \(d\).

• \(S_{4}=\frac{4}{2}(2u_1+3d)=14.5\), so \(4u_1+6d=14.5\)
• \(S_{9}=\frac{9}{2}(2u_1+8d)=49.5\), so \(9u_1+36d=49.5\)
• Simultaneous equation solver: u1 = 2.50, d = 0.750