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Add the following algebraic expressions.

Mathematics

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16 steps

Here are the solutions to the addition problems:

1.

  1. Write the expression for addition.

    8ab+(−5ab)+3ab+(−ab)8ab + (-5ab) + 3ab + (-ab)

  2. Combine the coefficients of the like terms.

    (8−5+3−1)ab=(3+3−1)ab=(6−1)ab=5ab(8 - 5 + 3 - 1)ab = (3 + 3 - 1)ab = (6 - 1)ab = 5ab

    The sum is 5ab\boxed{5ab}.

    2.

  3. Write the expression for addition.

    7x+(−3x)+5x+(−x)+(−2x)7x + (-3x) + 5x + (-x) + (-2x)

  4. Combine the coefficients of the like terms.

    (7−3+5−1−2)x=(4+5−1−2)x=(9−1−2)x=(8−2)x=6x(7 - 3 + 5 - 1 - 2)x = (4 + 5 - 1 - 2)x = (9 - 1 - 2)x = (8 - 2)x = 6x

    The sum is 6x\boxed{6x}.

    3.

  5. Write the expression for addition, grouping like terms.

    (3a−4b+4c)+(2a+3b−8c)+(a−6b+c)(3a - 4b + 4c) + (2a + 3b - 8c) + (a - 6b + c) =(3a+2a+a)+(−4b+3b−6b)+(4c−8c+c)= (3a + 2a + a) + (-4b + 3b - 6b) + (4c - 8c + c)

  6. Combine the coefficients for each variable.

    =(3+2+1)a+(−4+3−6)b+(4−8+1)c= (3 + 2 + 1)a + (-4 + 3 - 6)b + (4 - 8 + 1)c =6a+(−1−6)b+(−4+1)c= 6a + (-1 - 6)b + (-4 + 1)c =6a−7b−3c= 6a - 7b - 3c

    The sum is 6a−7b−3c\boxed{6a - 7b - 3c}.

    4.

  7. Write the expression for addition, grouping like terms.

    (5x−8y+2z)+(3z−4y−2x)+(6y−z−x)+(3x−2z−3y)(5x - 8y + 2z) + (3z - 4y - 2x) + (6y - z - x) + (3x - 2z - 3y) =(5x−2x−x+3x)+(−8y−4y+6y−3y)+(2z+3z−z−2z)= (5x - 2x - x + 3x) + (-8y - 4y + 6y - 3y) + (2z + 3z - z - 2z)

  8. Combine the coefficients for each variable.

    =(5−2−1+3)x+(−8−4+6−3)y+(2+3−1−2)z= (5 - 2 - 1 + 3)x + (-8 - 4 + 6 - 3)y + (2 + 3 - 1 - 2)z =(3−1+3)x+(−12+6−3)y+(5−1−2)z= (3 - 1 + 3)x + (-12 + 6 - 3)y + (5 - 1 - 2)z =(2+3)x+(−6−3)y+(4−2)z= (2 + 3)x + (-6 - 3)y + (4 - 2)z =5x−9y+2z= 5x - 9y + 2z

    The sum is 5x−9y+2z\boxed{5x - 9y + 2z}.

    5.

  9. Write the expression for addition, grouping like terms.

    (6ax−2by+3cz)+(6by−11ax−cz)+(10cz−2ax−3by)(6ax - 2by + 3cz) + (6by - 11ax - cz) + (10cz - 2ax - 3by) =(6ax−11ax−2ax)+(−2by+6by−3by)+(3cz−cz+10cz)= (6ax - 11ax - 2ax) + (-2by + 6by - 3by) + (3cz - cz + 10cz)

  10. Combine the coefficients for each variable.

    =(6−11−2)ax+(−2+6−3)by+(3−1+10)cz= (6 - 11 - 2)ax + (-2 + 6 - 3)by + (3 - 1 + 10)cz =(−5−2)ax+(4−3)by+(2+10)cz= (-5 - 2)ax + (4 - 3)by + (2 + 10)cz =−7ax+by+12cz= -7ax + by + 12cz

    The sum is −7ax+by+12cz\boxed{-7ax + by + 12cz}.

    6.

  11. Write the expression for addition, grouping like terms by degree.

    (2x3−9x2+8)+(3x2−6x−5)+(7x3−10x+1)+(3+2x−5x2−4x3)(2x^3 - 9x^2 + 8) + (3x^2 - 6x - 5) + (7x^3 - 10x + 1) + (3 + 2x - 5x^2 - 4x^3) =(2x3+7x3−4x3)+(−9x2+3x2−5x2)+(−6x−10x+2x)+(8−5+1+3)= (2x^3 + 7x^3 - 4x^3) + (-9x^2 + 3x^2 - 5x^2) + (-6x - 10x + 2x) + (8 - 5 + 1 + 3)

  12. Combine the coefficients for each term.

    =(2+7−4)x3+(−9+3−5)x2+(−6−10+2)x+(8−5+1+3)= (2 + 7 - 4)x^3 + (-9 + 3 - 5)x^2 + (-6 - 10 + 2)x + (8 - 5 + 1 + 3) =(9−4)x3+(−6−5)x2+(−16+2)x+(3+1+3)= (9 - 4)x^3 + (-6 - 5)x^2 + (-16 + 2)x + (3 + 1 + 3) =5x3−11x2−14x+7= 5x^3 - 11x^2 - 14x + 7

    The sum is 5x3−11x2−14x+7\boxed{5x^3 - 11x^2 - 14x + 7}.

    7.

  13. Write the expression for addition, grouping like terms.

    (6p+4q−r+3)+(2r−5p−6)+(11q−7p+2r−1)+(2q−3r+4)(6p + 4q - r + 3) + (2r - 5p - 6) + (11q - 7p + 2r - 1) + (2q - 3r + 4) =(6p−5p−7p)+(4q+11q+2q)+(−r+2r+2r−3r)+(3−6−1+4)= (6p - 5p - 7p) + (4q + 11q + 2q) + (-r + 2r + 2r - 3r) + (3 - 6 - 1 + 4)

  14. Combine the coefficients for each variable and constant.

    =(6−5−7)p+(4+11+2)q+(−1+2+2−3)r+(3−6−1+4)= (6 - 5 - 7)p + (4 + 11 + 2)q + (-1 + 2 + 2 - 3)r + (3 - 6 - 1 + 4) =(1−7)p+(15+2)q+(1+2−3)r+(−3−1+4)= (1 - 7)p + (15 + 2)q + (1 + 2 - 3)r + (-3 - 1 + 4) =−6p+17q+(3−3)r+(−4+4)= -6p + 17q + (3 - 3)r + (-4 + 4) =−6p+17q+0r+0= -6p + 17q + 0r + 0 =−6p+17q= -6p + 17q

    The sum is −6p+17q\boxed{-6p + 17q}.

    8.

  15. Write the expression for addition, grouping like terms.

    (4x2−7xy+4y2−3)+(5+6y2−8xy+x2)+(6−2xy+2x2−5y2)(4x^2 - 7xy + 4y^2 - 3) + (5 + 6y^2 - 8xy + x^2) + (6 - 2xy + 2x^2 - 5y^2) =(4x2+x2+2x2)+(−7xy−8xy−2xy)+(4y2+6y2−5y2)+(−3+5+6)= (4x^2 + x^2 + 2x^2) + (-7xy - 8xy - 2xy) + (4y^2 + 6y^2 - 5y^2) + (-3 + 5 + 6)

  16. Combine the coefficients for each term.

    =(4+1+2)x2+(−7−8−2)xy+(4+6−5)y2+(−3+5+6)= (4 + 1 + 2)x^2 + (-7 - 8 - 2)xy + (4 + 6 - 5)y^2 + (-3 + 5 + 6) =7x2+(−15−2)xy+(10−5)y2+(2+6)= 7x^2 + (-15 - 2)xy + (10 - 5)y^2 + (2 + 6) =7x2−17xy+5y2+8= 7x^2 - 17xy + 5y^2 + 8

    The sum is 7x2−17xy+5y2+8\boxed{7x^2 - 17xy + 5y^2 + 8}.

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