{"id":14227,"date":"2018-06-15T19:27:44","date_gmt":"2018-06-15T19:27:44","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/chapter\/section-exercises-37\/"},"modified":"2022-01-09T20:13:06","modified_gmt":"2022-01-09T20:13:06","slug":"problem-set-cramers-rule","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/problem-set-cramers-rule\/","title":{"raw":"Problem Set: Solving Systems with Cramer's Rule","rendered":"Problem Set: Solving Systems with Cramer’s Rule"},"content":{"raw":"1. Explain why we can always evaluate the determinant of a square matrix.\r\n\r\n2.\u00a0Examining Cramer\u2019s Rule, explain why there is no unique solution to the system when the determinant of your matrix is 0. For simplicity, use a [latex]2\\times 2[\/latex] matrix.\r\n\r\n3. Explain what it means in terms of an inverse for a matrix to have a 0 determinant.\r\n\r\n4.\u00a0The determinant of [latex]2\\times 2[\/latex] matrix [latex]A[\/latex] is 3. If you switch the rows and multiply the first row by 6 and the second row by 2, explain how to find the determinant and provide the answer.\r\n\r\nFor the following exercises, find the determinant.\r\n\r\n5. [latex]|\\begin{array}{cc}1& 2\\\\ 3& 4\\end{array}|[\/latex]\r\n\r\n6.\u00a0[latex]|\\begin{array}{rr}\\hfill -1& \\hfill 2\\\\ \\hfill 3& \\hfill -4\\end{array}|[\/latex]\r\n\r\n7. [latex]|\\begin{array}{rr}\\hfill 2& \\hfill -5\\\\ \\hfill -1& \\hfill 6\\end{array}|[\/latex]\r\n\r\n8.\u00a0[latex]|\\begin{array}{cc}-8& 4\\\\ -1& 5\\end{array}|[\/latex]\r\n\r\n9. [latex]|\\begin{array}{rr}\\hfill 1& \\hfill 0\\\\ \\hfill 3& \\hfill -4\\end{array}|[\/latex]\r\n\r\n10.\u00a0[latex]|\\begin{array}{rr}\\hfill 10& \\hfill 20\\\\ \\hfill 0& \\hfill -10\\end{array}|[\/latex]\r\n\r\n11. [latex]|\\begin{array}{cc}10& 0.2\\\\ 5& 0.1\\end{array}|[\/latex]\r\n\r\n12.\u00a0[latex]|\\begin{array}{rr}\\hfill 6& \\hfill -3\\\\ \\hfill 8& \\hfill 4\\end{array}|[\/latex]\r\n\r\n13. [latex]|\\begin{array}{rr}\\hfill -2& \\hfill -3\\\\ \\hfill 3.1& \\hfill 4,000\\end{array}|[\/latex]\r\n\r\n14.\u00a0[latex]|\\begin{array}{rr}\\hfill -1.1& \\hfill 0.6\\\\ \\hfill 7.2& \\hfill -0.5\\end{array}|[\/latex]\r\n\r\n15. [latex]|\\begin{array}{rrr}\\hfill -1& \\hfill 0& \\hfill 0\\\\ \\hfill 0& \\hfill 1& \\hfill 0\\\\ \\hfill 0& \\hfill 0& \\hfill -3\\end{array}|[\/latex]\r\n\r\n16.\u00a0[latex]|\\begin{array}{rrr}\\hfill -1& \\hfill 4& \\hfill 0\\\\ \\hfill 0& \\hfill 2& \\hfill 3\\\\ \\hfill 0& \\hfill 0& \\hfill -3\\end{array}|[\/latex]\r\n\r\n17. [latex]|\\begin{array}{ccc}1& 0& 1\\\\ 0& 1& 0\\\\ 1& 0& 0\\end{array}|[\/latex]\r\n\r\n18.\u00a0[latex]|\\begin{array}{rrr}\\hfill 2& \\hfill -3& \\hfill 1\\\\ \\hfill 3& \\hfill -4& \\hfill 1\\\\ \\hfill -5& \\hfill 6& \\hfill 1\\end{array}|[\/latex]\r\n\r\n19. [latex]|\\begin{array}{rrr}\\hfill -2& \\hfill 1& \\hfill 4\\\\ \\hfill -4& \\hfill 2& \\hfill -8\\\\ \\hfill 2& \\hfill -8& \\hfill -3\\end{array}|[\/latex]\r\n\r\n20.\u00a0[latex]|\\begin{array}{rrr}\\hfill 6& \\hfill -1& \\hfill 2\\\\ \\hfill -4& \\hfill -3& \\hfill 5\\\\ \\hfill 1& \\hfill 9& \\hfill -1\\end{array}|[\/latex]\r\n\r\n21. [latex]|\\begin{array}{rrr}\\hfill 5& \\hfill 1& \\hfill -1\\\\ \\hfill 2& \\hfill 3& \\hfill 1\\\\ \\hfill 3& \\hfill -6& \\hfill -3\\end{array}|[\/latex]\r\n\r\n22.\u00a0[latex]|\\begin{array}{rrr}\\hfill 1.1& \\hfill 2& \\hfill -1\\\\ \\hfill -4& \\hfill 0& \\hfill 0\\\\ \\hfill 4.1& \\hfill -0.4& \\hfill 2.5\\end{array}|[\/latex]\r\n\r\n23. [latex]|\\begin{array}{rrr}\\hfill 2& \\hfill -1.6& \\hfill 3.1\\\\ \\hfill 1.1& \\hfill 3& \\hfill -8\\\\ \\hfill -9.3& \\hfill 0& \\hfill 2\\end{array}|[\/latex]\r\n\r\n24.\u00a0[latex]|\\begin{array}{ccc}-\\frac{1}{2}& \\frac{1}{3}& \\frac{1}{4}\\\\ \\frac{1}{5}& -\\frac{1}{6}& \\frac{1}{7}\\\\ 0& 0& \\frac{1}{8}\\end{array}|[\/latex]\r\n\r\nFor the following exercises, solve the system of linear equations using Cramer\u2019s Rule.\r\n\r\n25. [latex]\\begin{array}{l}2x - 3y=-1\\\\ 4x+5y=9\\end{array}[\/latex]\r\n\r\n26.\u00a0[latex]\\begin{array}{r}5x - 4y=2\\\\ -4x+7y=6\\end{array}[\/latex]\r\n\r\n27. [latex]\\begin{array}{l}\\text{ }6x - 3y=2\\hfill \\\\ -8x+9y=-1\\hfill \\end{array}[\/latex]\r\n\r\n28.\u00a0[latex]\\begin{array}{l}2x+6y=12\\\\ 5x - 2y=13\\end{array}[\/latex]\r\n\r\n29. [latex]\\begin{array}{l}4x+3y=23\\hfill \\\\ \\text{ }2x-y=-1\\hfill \\end{array}[\/latex]\r\n\r\n30.\u00a0[latex]\\begin{array}{l}10x - 6y=2\\hfill \\\\ -5x+8y=-1\\hfill \\end{array}[\/latex]\r\n\r\n31. [latex]\\begin{array}{l}4x - 3y=-3\\\\ 2x+6y=-4\\end{array}[\/latex]\r\n\r\n32.\u00a0[latex]\\begin{array}{r}4x - 5y=7\\\\ -3x+9y=0\\end{array}[\/latex]\r\n\r\n33. [latex]\\begin{array}{l}4x+10y=180\\hfill \\\\ -3x - 5y=-105\\hfill \\end{array}[\/latex]\r\n\r\n34.\u00a0[latex]\\begin{array}{l}\\text{ }8x - 2y=-3\\hfill \\\\ -4x+6y=4\\hfill \\end{array}[\/latex]\r\n\r\nFor the following exercises, solve the system of linear equations using Cramer\u2019s Rule.\r\n\r\n35. [latex]\\begin{array}{l}\\text{ }x+2y - 4z=-1\\hfill \\\\ \\text{ }7x+3y+5z=26\\hfill \\\\ -2x - 6y+7z=-6\\hfill \\end{array}[\/latex]\r\n\r\n36.\u00a0[latex]\\begin{array}{l}-5x+2y - 4z=-47\\hfill \\\\ \\text{ }4x - 3y-z=-94\\hfill \\\\ \\text{ }3x - 3y+2z=94\\hfill \\end{array}[\/latex]\r\n\r\n37. [latex]\\begin{array}{l}\\text{ }4x+5y-z=-7\\hfill \\\\ -2x - 9y+2z=8\\hfill \\\\ \\text{ }5y+7z=21\\hfill \\end{array}[\/latex]\r\n\r\n38.\u00a0[latex]\\begin{array}{r}4x - 3y+4z=10\\\\ 5x - 2z=-2\\\\ 3x+2y - 5z=-9\\end{array}[\/latex]\r\n\r\n39. [latex]\\begin{array}{l}4x - 2y+3z=6\\hfill \\\\ \\text{ }-6x+y=-2\\hfill \\\\ 2x+7y+8z=24\\hfill \\end{array}[\/latex]\r\n\r\n40.\u00a0[latex]\\begin{array}{r}\\hfill 5x+2y-z=1\\\\ \\hfill -7x - 8y+3z=1.5\\\\ \\hfill 6x - 12y+z=7\\end{array}[\/latex]\r\n\r\n41. [latex]\\begin{array}{l}\\text{ }13x - 17y+16z=73\\hfill \\\\ -11x+15y+17z=61\\hfill \\\\ \\text{ }46x+10y - 30z=-18\\hfill \\end{array}[\/latex]\r\n\r\n42.\u00a0[latex]\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ -4x - 3y - 8z=-7\\hfill \\end{array}\\hfill \\\\ \\text{ }2x - 9y+5z=0.5\\hfill \\\\ \\text{ }5x - 6y - 5z=-2\\hfill \\end{array}[\/latex]\r\n\r\n43. [latex]\\begin{array}{l}\\text{ }4x - 6y+8z=10\\hfill \\\\ -2x+3y - 4z=-5\\hfill \\\\ \\text{ }x+y+z=1\\hfill \\end{array}[\/latex]\r\n\r\n44.\u00a0[latex]\\begin{array}{r}\\hfill 4x - 6y+8z=10\\\\ \\hfill -2x+3y - 4z=-5\\\\ \\hfill 12x+18y - 24z=-30\\end{array}[\/latex]\r\n\r\nFor the following exercises, use the determinant function on a graphing utility.\r\n\r\n45. [latex]|\\begin{array}{rrrr}\\hfill 1& \\hfill 0& \\hfill 8& \\hfill 9\\\\ \\hfill 0& \\hfill 2& \\hfill 1& \\hfill 0\\\\ \\hfill 1& \\hfill 0& \\hfill 3& \\hfill 0\\\\ \\hfill 0& \\hfill 2& \\hfill 4& \\hfill 3\\end{array}|[\/latex]\r\n\r\n46.\u00a0[latex]|\\begin{array}{rrrr}\\hfill 1& \\hfill 0& \\hfill 2& \\hfill 1\\\\ \\hfill 0& \\hfill -9& \\hfill 1& \\hfill 3\\\\ \\hfill 3& \\hfill 0& \\hfill -2& \\hfill -1\\\\ \\hfill 0& \\hfill 1& \\hfill 1& \\hfill -2\\end{array}|[\/latex]\r\n\r\n47. [latex]|\\begin{array}{rrrr}\\hfill \\frac{1}{2}& \\hfill 1& \\hfill 7& \\hfill 4\\\\ \\hfill 0& \\hfill \\frac{1}{2}& \\hfill 100& \\hfill 5\\\\ \\hfill 0& \\hfill 0& \\hfill 2& \\hfill 2,000\\\\ \\hfill 0& \\hfill 0& \\hfill 0& \\hfill 2\\end{array}|[\/latex]\r\n\r\n48.\u00a0[latex]|\\begin{array}{rrrr}\\hfill 1& \\hfill 0& \\hfill 0& \\hfill 0\\\\ \\hfill 2& \\hfill 3& \\hfill 0& \\hfill 0\\\\ \\hfill 4& \\hfill 5& \\hfill 6& \\hfill 0\\\\ \\hfill 7& \\hfill 8& \\hfill 9& \\hfill 0\\end{array}|[\/latex]","rendered":"

1. Explain why we can always evaluate the determinant of a square matrix.<\/p>\n

2.\u00a0Examining Cramer\u2019s Rule, explain why there is no unique solution to the system when the determinant of your matrix is 0. For simplicity, use a [latex]2\\times 2[\/latex] matrix.<\/p>\n

3. Explain what it means in terms of an inverse for a matrix to have a 0 determinant.<\/p>\n

4.\u00a0The determinant of [latex]2\\times 2[\/latex] matrix [latex]A[\/latex] is 3. If you switch the rows and multiply the first row by 6 and the second row by 2, explain how to find the determinant and provide the answer.<\/p>\n

For the following exercises, find the determinant.<\/p>\n

5. [latex]|\\begin{array}{cc}1& 2\\\\ 3& 4\\end{array}|[\/latex]<\/p>\n

6.\u00a0[latex]|\\begin{array}{rr}\\hfill -1& \\hfill 2\\\\ \\hfill 3& \\hfill -4\\end{array}|[\/latex]<\/p>\n

7. [latex]|\\begin{array}{rr}\\hfill 2& \\hfill -5\\\\ \\hfill -1& \\hfill 6\\end{array}|[\/latex]<\/p>\n

8.\u00a0[latex]|\\begin{array}{cc}-8& 4\\\\ -1& 5\\end{array}|[\/latex]<\/p>\n

9. [latex]|\\begin{array}{rr}\\hfill 1& \\hfill 0\\\\ \\hfill 3& \\hfill -4\\end{array}|[\/latex]<\/p>\n

10.\u00a0[latex]|\\begin{array}{rr}\\hfill 10& \\hfill 20\\\\ \\hfill 0& \\hfill -10\\end{array}|[\/latex]<\/p>\n

11. [latex]|\\begin{array}{cc}10& 0.2\\\\ 5& 0.1\\end{array}|[\/latex]<\/p>\n

12.\u00a0[latex]|\\begin{array}{rr}\\hfill 6& \\hfill -3\\\\ \\hfill 8& \\hfill 4\\end{array}|[\/latex]<\/p>\n

13. [latex]|\\begin{array}{rr}\\hfill -2& \\hfill -3\\\\ \\hfill 3.1& \\hfill 4,000\\end{array}|[\/latex]<\/p>\n

14.\u00a0[latex]|\\begin{array}{rr}\\hfill -1.1& \\hfill 0.6\\\\ \\hfill 7.2& \\hfill -0.5\\end{array}|[\/latex]<\/p>\n

15. [latex]|\\begin{array}{rrr}\\hfill -1& \\hfill 0& \\hfill 0\\\\ \\hfill 0& \\hfill 1& \\hfill 0\\\\ \\hfill 0& \\hfill 0& \\hfill -3\\end{array}|[\/latex]<\/p>\n

16.\u00a0[latex]|\\begin{array}{rrr}\\hfill -1& \\hfill 4& \\hfill 0\\\\ \\hfill 0& \\hfill 2& \\hfill 3\\\\ \\hfill 0& \\hfill 0& \\hfill -3\\end{array}|[\/latex]<\/p>\n

17. [latex]|\\begin{array}{ccc}1& 0& 1\\\\ 0& 1& 0\\\\ 1& 0& 0\\end{array}|[\/latex]<\/p>\n

18.\u00a0[latex]|\\begin{array}{rrr}\\hfill 2& \\hfill -3& \\hfill 1\\\\ \\hfill 3& \\hfill -4& \\hfill 1\\\\ \\hfill -5& \\hfill 6& \\hfill 1\\end{array}|[\/latex]<\/p>\n

19. [latex]|\\begin{array}{rrr}\\hfill -2& \\hfill 1& \\hfill 4\\\\ \\hfill -4& \\hfill 2& \\hfill -8\\\\ \\hfill 2& \\hfill -8& \\hfill -3\\end{array}|[\/latex]<\/p>\n

20.\u00a0[latex]|\\begin{array}{rrr}\\hfill 6& \\hfill -1& \\hfill 2\\\\ \\hfill -4& \\hfill -3& \\hfill 5\\\\ \\hfill 1& \\hfill 9& \\hfill -1\\end{array}|[\/latex]<\/p>\n

21. [latex]|\\begin{array}{rrr}\\hfill 5& \\hfill 1& \\hfill -1\\\\ \\hfill 2& \\hfill 3& \\hfill 1\\\\ \\hfill 3& \\hfill -6& \\hfill -3\\end{array}|[\/latex]<\/p>\n

22.\u00a0[latex]|\\begin{array}{rrr}\\hfill 1.1& \\hfill 2& \\hfill -1\\\\ \\hfill -4& \\hfill 0& \\hfill 0\\\\ \\hfill 4.1& \\hfill -0.4& \\hfill 2.5\\end{array}|[\/latex]<\/p>\n

23. [latex]|\\begin{array}{rrr}\\hfill 2& \\hfill -1.6& \\hfill 3.1\\\\ \\hfill 1.1& \\hfill 3& \\hfill -8\\\\ \\hfill -9.3& \\hfill 0& \\hfill 2\\end{array}|[\/latex]<\/p>\n

24.\u00a0[latex]|\\begin{array}{ccc}-\\frac{1}{2}& \\frac{1}{3}& \\frac{1}{4}\\\\ \\frac{1}{5}& -\\frac{1}{6}& \\frac{1}{7}\\\\ 0& 0& \\frac{1}{8}\\end{array}|[\/latex]<\/p>\n

For the following exercises, solve the system of linear equations using Cramer\u2019s Rule.<\/p>\n

25. [latex]\\begin{array}{l}2x - 3y=-1\\\\ 4x+5y=9\\end{array}[\/latex]<\/p>\n

26.\u00a0[latex]\\begin{array}{r}5x - 4y=2\\\\ -4x+7y=6\\end{array}[\/latex]<\/p>\n

27. [latex]\\begin{array}{l}\\text{ }6x - 3y=2\\hfill \\\\ -8x+9y=-1\\hfill \\end{array}[\/latex]<\/p>\n

28.\u00a0[latex]\\begin{array}{l}2x+6y=12\\\\ 5x - 2y=13\\end{array}[\/latex]<\/p>\n

29. [latex]\\begin{array}{l}4x+3y=23\\hfill \\\\ \\text{ }2x-y=-1\\hfill \\end{array}[\/latex]<\/p>\n

30.\u00a0[latex]\\begin{array}{l}10x - 6y=2\\hfill \\\\ -5x+8y=-1\\hfill \\end{array}[\/latex]<\/p>\n

31. [latex]\\begin{array}{l}4x - 3y=-3\\\\ 2x+6y=-4\\end{array}[\/latex]<\/p>\n

32.\u00a0[latex]\\begin{array}{r}4x - 5y=7\\\\ -3x+9y=0\\end{array}[\/latex]<\/p>\n

33. [latex]\\begin{array}{l}4x+10y=180\\hfill \\\\ -3x - 5y=-105\\hfill \\end{array}[\/latex]<\/p>\n

34.\u00a0[latex]\\begin{array}{l}\\text{ }8x - 2y=-3\\hfill \\\\ -4x+6y=4\\hfill \\end{array}[\/latex]<\/p>\n

For the following exercises, solve the system of linear equations using Cramer\u2019s Rule.<\/p>\n

35. [latex]\\begin{array}{l}\\text{ }x+2y - 4z=-1\\hfill \\\\ \\text{ }7x+3y+5z=26\\hfill \\\\ -2x - 6y+7z=-6\\hfill \\end{array}[\/latex]<\/p>\n

36.\u00a0[latex]\\begin{array}{l}-5x+2y - 4z=-47\\hfill \\\\ \\text{ }4x - 3y-z=-94\\hfill \\\\ \\text{ }3x - 3y+2z=94\\hfill \\end{array}[\/latex]<\/p>\n

37. [latex]\\begin{array}{l}\\text{ }4x+5y-z=-7\\hfill \\\\ -2x - 9y+2z=8\\hfill \\\\ \\text{ }5y+7z=21\\hfill \\end{array}[\/latex]<\/p>\n

38.\u00a0[latex]\\begin{array}{r}4x - 3y+4z=10\\\\ 5x - 2z=-2\\\\ 3x+2y - 5z=-9\\end{array}[\/latex]<\/p>\n

39. [latex]\\begin{array}{l}4x - 2y+3z=6\\hfill \\\\ \\text{ }-6x+y=-2\\hfill \\\\ 2x+7y+8z=24\\hfill \\end{array}[\/latex]<\/p>\n

40.\u00a0[latex]\\begin{array}{r}\\hfill 5x+2y-z=1\\\\ \\hfill -7x - 8y+3z=1.5\\\\ \\hfill 6x - 12y+z=7\\end{array}[\/latex]<\/p>\n

41. [latex]\\begin{array}{l}\\text{ }13x - 17y+16z=73\\hfill \\\\ -11x+15y+17z=61\\hfill \\\\ \\text{ }46x+10y - 30z=-18\\hfill \\end{array}[\/latex]<\/p>\n

42.\u00a0[latex]\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ -4x - 3y - 8z=-7\\hfill \\end{array}\\hfill \\\\ \\text{ }2x - 9y+5z=0.5\\hfill \\\\ \\text{ }5x - 6y - 5z=-2\\hfill \\end{array}[\/latex]<\/p>\n

43. [latex]\\begin{array}{l}\\text{ }4x - 6y+8z=10\\hfill \\\\ -2x+3y - 4z=-5\\hfill \\\\ \\text{ }x+y+z=1\\hfill \\end{array}[\/latex]<\/p>\n

44.\u00a0[latex]\\begin{array}{r}\\hfill 4x - 6y+8z=10\\\\ \\hfill -2x+3y - 4z=-5\\\\ \\hfill 12x+18y - 24z=-30\\end{array}[\/latex]<\/p>\n

For the following exercises, use the determinant function on a graphing utility.<\/p>\n

45. [latex]|\\begin{array}{rrrr}\\hfill 1& \\hfill 0& \\hfill 8& \\hfill 9\\\\ \\hfill 0& \\hfill 2& \\hfill 1& \\hfill 0\\\\ \\hfill 1& \\hfill 0& \\hfill 3& \\hfill 0\\\\ \\hfill 0& \\hfill 2& \\hfill 4& \\hfill 3\\end{array}|[\/latex]<\/p>\n

46.\u00a0[latex]|\\begin{array}{rrrr}\\hfill 1& \\hfill 0& \\hfill 2& \\hfill 1\\\\ \\hfill 0& \\hfill -9& \\hfill 1& \\hfill 3\\\\ \\hfill 3& \\hfill 0& \\hfill -2& \\hfill -1\\\\ \\hfill 0& \\hfill 1& \\hfill 1& \\hfill -2\\end{array}|[\/latex]<\/p>\n

47. [latex]|\\begin{array}{rrrr}\\hfill \\frac{1}{2}& \\hfill 1& \\hfill 7& \\hfill 4\\\\ \\hfill 0& \\hfill \\frac{1}{2}& \\hfill 100& \\hfill 5\\\\ \\hfill 0& \\hfill 0& \\hfill 2& \\hfill 2,000\\\\ \\hfill 0& \\hfill 0& \\hfill 0& \\hfill 2\\end{array}|[\/latex]<\/p>\n

48.\u00a0[latex]|\\begin{array}{rrrr}\\hfill 1& \\hfill 0& \\hfill 0& \\hfill 0\\\\ \\hfill 2& \\hfill 3& \\hfill 0& \\hfill 0\\\\ \\hfill 4& \\hfill 5& \\hfill 6& \\hfill 0\\\\ \\hfill 7& \\hfill 8& \\hfill 9& \\hfill 0\\end{array}|[\/latex]<\/p>\n\n\t\t\t

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Candela Citations<\/h3>\n\t\t\t\t\t
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CC licensed content, Specific attribution<\/div>