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Systems Of Linear Equations

Prep for Systems of Linear Equations

      1. [latex]x=8[/latex]
      2. one
      1. answers may vary
      2. infinitely many
      1. [latex](2,3)[/latex]
      2. one
    1. \begin{array}{l}\text{B}\hspace{1em}x+y=6\\\text{D}\hspace{1em}3x-4y=1\\\text{C}\hspace{1em}-y=3x-9\\\text{A}\hspace{1em}y=2x-7\end{array}
    1. [latex]21x - 14y = 56[/latex]
    2. [latex]6x - y = 27[/latex]
    3. [latex]-4x + 10 = 40[/latex]
    4. [latex]-\frac{6}{7}x = -2y - 16[/latex]
    1. [latex]y = \frac{3}{2}x + 4[/latex]
    2. [latex]y = -x + 3[/latex]
    3. [latex]y = -\frac{2}{3}x + 5[/latex]
    4. [latex]y = x - 3[/latex]
    5. [latex]y = -x + 4[/latex]
    6. [latex]y = -2x - 1[/latex]
    7. [latex]y = \frac{3}{4}x - 6[/latex]
    8. [latex]y = \frac{4}{5}x - 10[/latex]
    9. [latex]y=\frac{-1}4x+3[/latex]
    1. x = the number of fill-in-the-blank questionsy = the number of multiple choice questions
    2. x = the number of grams of medicine Ay = the number of grams of medicine B
    3. x = the cost of a small boxy = the cost of a large box
    4. x = the number of childreny = the number of adults

Systems of Linear Equations

  1. No, the given point does not satisfy the second equation.
    1. [latex](3,6)[/latex]
    2. [latex](-1,3)[/latex]
    3. No solution
    4. [latex]\left(\frac47,\frac{27}7\right)[/latex]
    1. 45 cats and 60 dogs
    2. 203 adults and 226 children
    3. 15 tens and 6 fives
    4. $40,000 at 0.50% and 90,000 at 1.1%

Matrices

  1. [latex]A\;=\;3\times2,\;B\;=\;3\times2,\;C\;=\;2\times3,\;D\;=\;2\times1[/latex]
    1. [latex]\begin{bmatrix}5&-4&9\\1&-2&-3\end{bmatrix}[/latex]
    2. [latex]\begin{bmatrix}3&-2&-3&-5\\1&2&3&-3\\-1&4&-6&-1\end{bmatrix}[/latex]
  3. [latex]\left\{\begin{array}{l}3x-2y+4z=6\\9x+4y-z=0\\6x-8y-3z=3\end{array}\right.[/latex]
  4. [latex]\left(-7,4\right)[/latex]

Homework: Systems of Linear Equations

    1. [latex]4\times2[/latex]
    1. [latex](0, 8)[/latex]
    2. No Solution
    3. [latex](4, 6)[/latex]
    4. [latex](-1, 2)[/latex]
    5. [latex](-1, 8)[/latex]
    6. [latex]\left(-\frac{65}{4}, \frac{35}{16}\right)[/latex]
    7. [latex](-9, 3)[/latex]
    8. [latex]\left(\frac{33}{23}, \frac{96}{23}\right)[/latex]
    1. Sour Sillies: $2.00, Choco Nuts: 1.50
    2. 14 two-pointers and 8 free throws
    3. 12 pigs and 5 chickens
    4. 100 children and 35 adults
    5. tree: $75.00, bush: $25.00
    6. 15 true/false and 5 multiple choice
    7. 30 g of medicine A and 6 grams of medicine B
    8. $13.00

Fractions

    1. Check with a classmate.
    2. Many answers are possible.
    3. Compare with a neighbor.
    1. [latex]\frac{1}{12}[/latex]
    2. [latex]\frac{5}{16}[/latex]
    3. [latex]\frac{7}{9}[/latex]
    4. [latex]\frac{4}{21}[/latex]
    5. [latex]\frac{1}{10}[/latex]
    6. [latex]\frac{52}{81}[/latex]
    7. [latex]\frac{21}{40}[/latex]
    8. [latex]7 \frac{1}{10}[/latex]
    9. [latex]\frac{7}{18}[/latex]
    10. [latex]\frac{8}{9}[/latex]
    1. Reduce to lowest terms.
    2. Find the LCD.
    3. Find the LCD.
    4. Reciprocal mistake.

Applications with Fractions

    1. 8 lbs
    2. 8 teams
    3. [latex]33\frac14\text{lbs}[/latex]
    4. [latex]12\frac58\text{mpg}[/latex]
    5. 5 lbs
    6. [latex]5\frac15\text{laps}[/latex]

Homework: Fractions

    1. [latex]\frac{3}{4}[/latex]
    2. [latex]8 \frac{1}{4}[/latex]
    3. [latex]\frac{8}{9}[/latex]
    4. [latex]5 \frac{8}{15}[/latex]
    5. [latex]\frac{6}{35}[/latex]
    6. [latex]4 \frac{5}{9}[/latex]
    7. [latex]1 \frac{7}{24}[/latex]
    8. [latex]3 \frac{10}{21}[/latex]
    9. [latex]2 \frac{14}{27}[/latex]
    10. [latex]16 \frac{25}{36}[/latex]
    11. [latex]\frac{11}{18}[/latex]
    12. [latex]2 \frac{10}{27}[/latex]
    13. [latex]6 \frac{1}{3}[/latex]
    14. [latex]3 \frac{13}{20}[/latex]
    15. [latex]6 \frac{3}{5}[/latex]
    16. [latex]2 \frac{1}{8}[/latex]
    1. Find the LCD.
    1. [latex]1 \frac{7}{8}[/latex]
    2. [latex]7 \frac{7}{12}[/latex]
    3. [latex]\frac{7}{10}[/latex]
    4. [latex]\frac{1}{6}[/latex]
    5. [latex]\frac{8}{9}[/latex] servings
    6. 6 cupcakes, [latex]\frac{1}{6}[/latex] oz left
    7. [latex]14 \frac{8}{15}[/latex] mi
    8. [latex]1 \frac{1}{2}[/latex] qts
    9. [latex]8 \frac{3}{4}[/latex] bags

Order of Operations

    1. [latex]578[/latex]
    2. [latex]-104[/latex]
    3. [latex]4[/latex]
    4. [latex]13[/latex]
    5. [latex]25 \frac{1}{14}[/latex]
    6. [latex]1 \frac{11}{120}[/latex]
    7. [latex]-108[/latex]
    8. [latex]21[/latex]
    9. [latex]-7[/latex]
    10. [latex]\frac{7}{60}[/latex]
    11. [latex]4[/latex]
    12. [latex]-27[/latex]
    13. [latex]\frac{2}{3}[/latex]
    14. [latex]1 \frac{1}{15}[/latex]
    15. [latex]\frac{9}{2}[/latex]

Solving Linear Equations

    1. [latex]x = 0[/latex]
    2. [latex]x = \frac{4}{3}[/latex]
    3. [latex]x = -\frac{3}{2}[/latex]
    4. All real numbers
    5. [latex]t = 9[/latex]
    6. [latex]x = 1[/latex]
    7. [latex]x = \frac{1}{2}[/latex]
    8. [latex]x = \frac{35}{46}[/latex]
    9. No solution
    10. [latex]x = -16[/latex]
    11. [latex]x = -36[/latex]
    12. [latex]x = \frac{79}{77}[/latex]

Linear Expressions and Equations

    1. [latex]\frac{7}{12}x - 1[/latex]
    2. [latex]x = \frac{12}{7}[/latex]
    3. [latex]\frac{13}{4}x[/latex]
    4. [latex]x = 8[/latex]
    5. [latex]\frac{x-9}{2}[/latex]
    6. [latex]x = 9[/latex]

Homework: Order of Operations and Linear Equations

    1. [latex]1[/latex]
    2. [latex]1[/latex]
    3. [latex]75[/latex]
    4. [latex]\frac{1}{30}[/latex]
    5. [latex]2[/latex]
    6. [latex]15[/latex]
    7. [latex]-15[/latex]
    8. [latex]17[/latex]
    9. [latex]1 \frac{24}{25}[/latex]
    10. [latex]\frac{55}{216}[/latex]
    11. [latex]\frac{6}{7}[/latex]
    12. [latex]\frac{25}{96}[/latex]
    13. [latex]\frac{25}{78}[/latex]
    14. [latex]\frac{7}{12}[/latex]
    15. [latex]1 \frac{409}{1152}[/latex]
    1. [latex]x = 18[/latex]
    2. [latex]x = -3[/latex]
    3. [latex]x = 8[/latex]
    4. [latex]x = \frac{2}{3}[/latex]
    5. [latex]x = -6[/latex]
    6. [latex]x = 18[/latex]
    7. [latex]x = 6[/latex]
    8. [latex]x = -3[/latex]
    9. [latex]c = 12[/latex]
    10. [latex]x = 1[/latex]
    11. [latex]x = \frac{4}{3}[/latex]
    12. [latex]x = 2[/latex]
    13. [latex]x = \frac{5}{3}[/latex]
    14. [latex]x = -\frac{28}{5}[/latex]
    15. [latex]x = -\frac{28}{15}[/latex]
    16. [latex]x = -5[/latex]
    17. [latex]x = \frac{2}{15}[/latex]
    18. [latex]x = 0[/latex]
    19. [latex]x = 11[/latex]
    20. [latex]x = 3[/latex]
    21. [latex]x = 4[/latex]
    22. [latex]x = 6[/latex]
    23. [latex]x = 6[/latex]
    24. [latex]x = -22[/latex]
    25. [latex]x = -6[/latex]
    26. [latex]x = -\frac{5}{19}[/latex]
    27. [latex]x = -\frac{70}{19}[/latex]
    28. No solution

Operations with Polynomials

    1. [latex]6a^3 + 2a[/latex]
    2. [latex]-\frac{12}{5}a^3 + \frac{8}{5}a^2 + 9a[/latex]
    3. [latex]\frac{18}{5}a^3 + \frac{16}{7}a^2 - \frac{5}{7}a[/latex]
    4. [latex]16x^2 - 24x + 9[/latex]
    5. [latex]x^3 - 27[/latex]
    6. [latex]-r^2 - 12r + 25[/latex]

Applications with Polynomials

    1. [latex]5x^2 + 5x + 2[/latex]
    2. [latex]3x^2 - 20x + 25[/latex]
    3. [latex]10x + 10[/latex]
    4. [latex]x^2 + 42x + 360[/latex]

Greatest Common Factor

    1. [latex]6(y^2 + 1)[/latex]
    2. [latex]y^2(x - 3)[/latex]
    3. [latex]3x(x + 2)[/latex]
    4. [latex]4y^2(4x^2 - 7x + 12y^3)[/latex]
    5. [latex]5x^2y^3(5xy - 13y^2 - 16x^2)[/latex]

Homework: Operations with Polynomials, Applications, & GCF

    1. [latex]6x[/latex]
    2. [latex]4[/latex]
    3. [latex]9x^2 - 4[/latex]
    4. [latex]6x + 4[/latex]
    5. [latex]9x^2 + 12x + 4[/latex]
    6. [latex]12x^2 - x - 35[/latex]
    7. [latex]3x^2y + 3xy - 23[/latex]
    8. [latex]7x - 1[/latex]
    9. [latex]x^3 + 12x^2 + 3x + 4[/latex]
    10. [latex]7x - 1[/latex]
    11. [latex]18x^2 + 9x - 35[/latex]
    12. [latex]x^2 + 7x + 10[/latex]
    13. [latex]3x^2 + 2x + 1[/latex]
    14. [latex]x^2 + 6x + 9[/latex]
    15. [latex]3x^2 + 2x + 1[/latex]
    16. [latex]-x^2 - 2x + 10[/latex]
    1. [latex]4x^3 + 4x^2[/latex]
    2. [latex]2x^2 + 10x + 3[/latex]
    1. [latex]3x(2x + 5)[/latex]
    2. [latex]18x^3(2 + 5x^2 - 4x^4)[/latex]
    3. [latex]2x(x - 6)[/latex]
    4. [latex]2(7x^2 - 6x + 4)[/latex]
    5. [latex]2x(7x^2 - 6x + 4)[/latex]
    6. [latex]16x(2x - 1)[/latex]
    7. [latex]8(4x^2 - 2x + 1)[/latex]

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College Algebra for Non-STEM Majors Copyright © by Amy Collins Montalbano. All Rights Reserved.

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