Solving systems of equations Problem set B Worked solutions

Solve without a calculator. \(3x + 5y=17\) and \(4x - 3y=-16\).

• Label the equations (1) and (2).
• Multiply (1) by 3: \(9x + 15y=51\)
• Multiply (2) by 5: \(20x - 15y=-80\)
• Add the equations to remove y: \(29x=-29\)
• So \(x=-1\). Put this in (1): \(y=4\)

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

1. (a) \((4x-1)(x+6)=15\)

   • Expand the brackets: \(4x^2 + 23x - 6=15\)
   • Subtract 15 from both sides: \(4x^2 + 23x - 21=0\)

2. (b) \(3x-\frac{8}{x}=5\)

   • Multiply every term by \(x\): \(3x^2-8=5x\)
   • Move \(5x\) to the left: \(3x^2 - 5x - 8=0\)

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

1. (a) \(7x + 3y=-5\), \(4x - 5y=14\)

   • Simultaneous equation solver: x = 0.362, y = −2.51
   • Graph check: the two lines meet at (0.362, −2.51)

2. (b) \(x=4y-3\), \(5x+2y=19\)

   • Standard form: \(x - 4y=-3\), \(5x + 2y=19\)
   • Simultaneous equation solver: x = 3.18, y = 1.55

Solve using your calculator. \(3x - y + 2z=-7\), \(x + 4y - 3z=7\), \(2x + y + z=0\).

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

Rearrange into standard form, then solve using your calculator. \(2x + y=3z + 6\), \(-y + 2z=-4x + 1\), \(x + z - 2=-3y - 4\).

• Standard form (variables on the left, number on the right): \(2x + y - 3z=6\), \(4x - y + 2z=1\), \(x + 3y + z=-2\)
• Simultaneous equation solver: x = 0.909, y = −0.455, z = −1.55
• Exact values: \(x=\frac{10}{11},\ y=-\frac{5}{11},\ z=-\frac{17}{11}\)

Solve using your calculator. \(2w + x + 3y - z=5\), \(w - 2x + y + 2z=-8\), \(-w + 3x + 2y + z=7\), \(3w + x - y + 2z=-6\).

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

Rearrange into standard form, then solve using your calculator. \(2w - x + 3y + z=6\), \(3w + x + 2z=y + 2\), \(w + 2x + 2y=3z - 5\), \(-2x + z=-2w - y + 4\).

• Standard form (variables on the left, number on the right): \(2w - x + 3y + z=6\), \(3w + x - y + 2z=2\), \(w + 2x + 2y - 3z=-5\), \(2w - 2x + y + z=4\)
• Simultaneous equation solver: w = −0.167, x = −0.487, y = 1.24, z = 2.12
• Exact values: \(w=-\frac{1}{6},\ x=-\frac{19}{39},\ y=\frac{97}{78},\ z=\frac{55}{26}\)

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

1. (a) \(x + 2y = 5\) and \(3x + 6y = 20\)

   • Multiply the first equation by 3: \(3x + 6y = 15\)
   • The second equation says \(3x + 6y = 20\). The lines are parallel: no solution.

2. (b) \(5x - 10y = 15\) and \(x - 2y = 3\)

   • Multiply the second equation by 5: \(5x - 10y = 15\)
   • This is the first equation. The lines are the same: infinitely many solutions.

3. (c) The system \(3x + ky = 7\) and \(12x + 20y = 5\) has no solution. Find \(k\).

   • Multiply the first equation by 4: \(12x + 4ky = 28\)
   • The left sides must match: \(4k = 20\), so \(k = 5\)
   • The constants differ (28 and 5), so there is no solution, as required.

A cafe sells coffees at one price and cakes at another. One table has 3 coffees and 2 cakes and pays $18.50. Another table has 5 coffees and 3 cakes and pays $29.45.

1. (a) Write two equations.

   • Let \(a\) = price of a coffee, \(b\) = price of a cake ($).
   • \(3a + 2b=18.5\)
   • \(5a + 3b=29.45\)

2. (b) Find the cost of 2 coffees and 4 cakes.

   • Simultaneous equation solver: a = 3.40, b = 4.15
   • 2a + 4b = 2 × 3.40 + 4 × 4.15 = $23.40

A saver's monthly deposits form an arithmetic sequence. The term \(u_{6}\) is 26 and the sum of the first 8 terms is 160. Find the first term \(u_1\) and the common difference \(d\).

• \(u_{6}=u_1+5d=26\)
• \(S_{8}=\frac{8}{2}(2u_1+7d)=160\), so \(8u_1+28d=160\)
• Simultaneous equation solver: u1 = 6, d = 4

A stack of logs is built layer by layer. The sum of the first 5 terms is 20 and the sum of the first 12 terms is 100.5. Find the first term \(u_1\) and the common difference \(d\).

• \(S_{5}=\frac{5}{2}(2u_1+4d)=20\), so \(5u_1+10d=20\)
• \(S_{12}=\frac{12}{2}(2u_1+11d)=100.5\), so \(12u_1+66d=100.5\)
• Simultaneous equation solver: u1 = 1.50, d = 1.25