Addition and Subtraction

Excerpts

Addition and Subtraction
If an algebraic expression contains only *integral forms*, that is, contains *no letter in the denominator of any of its terms*, it is called an **integral expression**.
Addition and Subtraction
Integral and fractional expressions are so named on account of the *form of the expressions*, and with no reference whatever to the *numerical value* of the expressions when definite numbers are put in place of the letters.
Addition and Subtraction
The coefficient of $x^{2}$ in the result will be $6 + 1$, or $7$; the coefficient of $x$ will be $-4 + 5$, or $1$; and the last term is $-5 + 4$, or $-1$.
Addition and Subtraction
When the coefficient of a term is $1$, it is not written, but understood.
Addition and Subtraction
This process is more easily performed by writing the subtrahend below the minuend, *mentally* changing the sign of each term in the subtrahend, and adding.
Addition and Subtraction
A parenthesis preceded by the sign $-$ may be removed, *provided the sign of every term within the parenthesis is changed*, namely, $+$ to $-$, and $-$ to $+$; and any number of terms may be enclosed within a parenthesis preceded by the sign $-$, *provided the sign of every term enclosed is changed*.
Addition and Subtraction
It is best to remove each parenthesis in succession, *beginning with the innermost*.

Equations

Addition and Subtraction
a + (b + c) &= a + b + c
A parenthesis preceded by + can be removed without changing the sign of any term inside it.
Addition and Subtraction
a + b + c &= a + (b + c)
Any group of terms can be enclosed in a parenthesis preceded by + without changing the sign of any term.
Addition and Subtraction
a + (b - c) &= a + b - c
A parenthesis preceded by + can be removed without changing the sign of the terms inside it, including a negative term.
Addition and Subtraction
a + b - c &= a + (b - c)
A group of terms including a negative term can be enclosed in a parenthesis preceded by + without changing any sign.
Addition and Subtraction
a - (b + c) &= a - b - c
A parenthesis preceded by - can be removed if the sign of every term inside it is changed.
Addition and Subtraction
a - b - c &= a - (b + c)
Terms can be enclosed in a parenthesis preceded by - provided the sign of every enclosed term is changed.
Addition and Subtraction
a - (b - c) &= a - b + c
A parenthesis preceded by - containing a negative term can be removed by changing every sign inside it.
Addition and Subtraction
a - b + c &= a - (b - c)
Terms can be enclosed in a parenthesis preceded by - by changing the sign of every enclosed term, here turning a negative term into a positive one.

Problems

Exercise 18

Exercise 18, problem 1, p. 48
$a^{2} - ab + b^{2}$; $a^{2} + ab + b^{2}$.
Printed answer:
- $2a^{2} + 2b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a**2 + 2*b**2
Exercise 18, problem 10, p. 48
$3y^{2} - x^{2} - 3xy$; $5x^{2} + 6xy - 7y^{2}$; $x^{2} + 2y^{2}$.
Printed answer:
- $5x^{2} + 3xy - 2y^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 5*x**2 + 3*x*y - 2*y**2
Exercise 18, problem 11, p. 48
$2a^{2} - 2ab + 3b^{2}$; $4b^{2} + 5ab - 2a^{2}$; $a^{2} - 3ab - 9b^{2}$.
Printed answer:
- $a^{2} - 2b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2 - 2*b**2
Exercise 18, problem 12, p. 48
$a^{3} - a^{2} + a - 1$; $a^{2} - 2a + 2$; $3a^{3} + 7a + 1$.
Printed answer:
- $4a^{3} + 6a + 2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*a**3 + 6*a + 2
Exercise 18, problem 13, p. 48
$2m^{3} - m^{2} - m$; $4m^{3} + 8m^{2} - 7$; $-3m^{3} + m + 9$.
Printed answer:
- $3m^{3} + 7m^{2} + 2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*m**3 + 7*m**2 + 2
Exercise 18, problem 14, p. 48
$x^{3} - 3x + 6y$; $x^{2} + 2x - 5y$; $x^{3} - 3x^{2} + 5x$.
Printed answer:
- $2x^{3} - 2x^{2} + 4x + y$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*x**3 - 2*x**2 + 4*x + y
Exercise 18, problem 15, p. 48
$6x^{3} - 5x + 1$; $x^{3} + 3x + 4$; $7x^{2} + 2x - 3$.
Printed answer:
- $7x^{3} + 7x^{2} + 2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 7*x**3 + 7*x**2 + 2
Exercise 18, problem 16, p. 48
$a^{3} + 3a^{2}b - 3ab^{2}$; $-3a^{2}b - 6ab^{2} - b^{3}$; $3a^{2}b + 4ab^{2}$.
Printed answer:
- $a^{3} + 3a^{2}b - 5ab^{2} - b^{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**3 + 3*a**2*b - 5*a*b**2 - b**3
Exercise 18, problem 17, p. 48
$a^{3} - 2a^{2}b - 2ab^{2}$; $a^{2}b - 3ab^{2} - b^{3}$; $3ab^{2} - 2a^{3} - b^{3}$.
Printed answer:
- $-a^{3} - a^{2}b - 2ab^{2} - 2b^{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -a**3 - a**2*b - 2*a*b**2 - 2*b**3
Exercise 18, problem 18, p. 48
$7x^{3} - 2x^{2}y + 9xy^{2} + 13y^{3}$; $5x^{2}y - 4xy^{2} - 2x^{3} - 3y^{3}$; $y^{3} - x^{3} - 3x^{2}y - 5xy^{2}$; $2x^{2}y - 5y^{3} - 2x^{3} - xy^{2}$.
Printed answer:
- $2x^{3} + 2x^{2}y - xy^{2} + 6y^{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*x**3 + 2*x**2*y - x*y**2 + 6*y**3
Exercise 18, problem 19, p. 48
Show that $x + y + z = 0$, if $x = a - b - c$, $y = 2b + 2c - 3a$, and $z = 2a - b - c$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 18, problem 2, p. 48
$3a^{2} + 5a-7$; $6a^{2} - 7a + 13$.
Printed answer:
- $9a^{2} - 2a + 6$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 9*a**2 - 2*a + 6
Exercise 18, problem 20, p. 48
Show that $x + y = 3z$, if $x = 3a^{2} - 6a + 12$, $y = 9a^{2} + 12a - 21$, and $z = 4a^{2} + 2a - 3$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 18, problem 3, p. 48
$x + 2y - 3z$; $-3x + y + 2z$; $2x - 3y + z$.
Printed answer:
- $0$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 0
Exercise 18, problem 4, p. 48
$3x + 2y - z$; $-x + 3y + 2z$; $2x - y + 3z$.
Printed answer:
- $4x + 4y + 4z$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*x + 4*y + 4*z
Exercise 18, problem 5, p. 48
$-3a + 2b + c$; $a - 3b + 2c$; $2a + 3b - c$.
Printed answer:
- $2b + 2c$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*b + 2*c
Exercise 18, problem 6, p. 48
$-a + 3b + 4c$; $3a - b + 2c$; $2a + 2b - 2c$.
Printed answer:
- $4a + 4b + 4c$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*a + 4*b + 4*c
Exercise 18, problem 7, p. 48
$4a^{2} + 3a + 5$; $-2a^{2} + 3a - 8$; $a^{2} - a + 1$.
Printed answer:
- $3a^{2} + 5a - 2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*a**2 + 5*a - 2
Exercise 18, problem 8, p. 48
$5ab + 6bc - 7ac$; $3ab - 9bc + 4ac$; $3bc + 6ac$.
Printed answer:
- $8ab + 3ac$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 8*a*b + 3*a*c
Exercise 18, problem 9, p. 48
$x^{3} + x^{2} + x$; $2x^{3} + 3x^{2} - 2x$; $3x^{3} - 4x^{2} + x$.
Printed answer:
- $6x^{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 6*x**3

Exercise 19

Exercise 19, problem 1, p. 50
$a - 2b + 3c$ from $2a - 3b + 4c$.
Printed answer:
- $a - b + c$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a - b + c
Exercise 19, problem 10, p. 50
$x^{4} + x - 5x^{3} + 5$ from $7 - 2x^{2} - 3x^{3} + x^{4}$.
Printed answer:
- $2 - x - 2x^{2} + 2x^{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2 - x - 2*x**2 + 2*x**3
Exercise 19, problem 11, p. 50
$a^{3} + b^{3} + c^{3} - 3abc$ from $3abc + a^{3} - 2b^{3} - 3c^{3}$.
Printed answer:
- $-3b^{3} - 4c^{3} + 6abc$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -3*b**3 - 4*c**3 + 6*a*b*c
Exercise 19, problem 12, p. 50
$2x^{4} - 5x^{2} + 7x - 3$ from $x^{4} + 2 - 2x^{3} - x^{2}$.
Printed answer:
- $-x^{4} - 2x^{3} + 4x^{2} - 7x + 5$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -x**4 - 2*x**3 + 4*x**2 - 7*x + 5
Exercise 19, problem 13, p. 50
$1 - x^{5} - x + x^{4} - x^{3}$ from $x^{4} + 1 + x + x^{2}$.
Printed answer:
- $2x + x^{2} + x^{3} + x^{5}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*x + x**2 + x**3 + x**5
Exercise 19, problem 14, p. 50
$a^{3} - b^{3} + 3a^{2}b - 3ab^{2}$ from $a^{3} + b^{3} - a^{2}b - ab^{2}$.
Printed answer:
- $2b^{3} - 4a^{2}b + 2ab^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*b**3 - 4*a**2*b + 2*a*b**2
Exercise 19, problem 15, p. 50
$a^{2} b - ab^{2} - 3a^{3} b^{3} - b^{4}$ from $b^{4} - 5a^{3} b^{3} - 2ab^{2} + a^{2} b$.
Printed answer:
- $2b^{4} - 2a^{3}b^{3} - ab^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*b**4 - 2*a**3*b**3 - a*b**2
Exercise 19, problem 16, p. 50
$-x^{3} + 7x^{2} y - 2y^{3} + 3xy^{2}$ from $3x^{3} + 5y^{3} - xy^{2} + 4x^{2}y$.
Printed answer:
- $4x^{3} - 3x^{2}y - 4xy^{2} + 7y^{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*x**3 - 3*x**2*y - 4*x*y**2 + 7*y**3
Exercise 19, problem 2, p. 50
$a - 3b - 5c$ from $3a - 5b + c$.
Printed answer:
- $2a - 2b + 6c$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a - 2*b + 6*c
Exercise 19, problem 3, p. 50
$2x - 4y + 6z$ from $4x - y - 2z$.
Printed answer:
- $2a + 3y -8z$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem 2*a + 3*y - 8*z
Exercise 19, problem 4, p. 50
$5x - 11y - 3z$ from $6x - 7y + 2z$.
Printed answer:
- $x + 4y + 5z$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x + 4*y + 5*z
Exercise 19, problem 5, p. 50
$ab - ac - bc + bd$ from $ab + ac + bc + bd$.
Printed answer:
- $2ac + 2bc$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a*c + 2*b*c
Exercise 19, problem 6, p. 50
$3ab + 2ac - 3bc + bd$ from $5ab - ac + bc + bd$.
Printed answer:
- $2ab - 3ac + 4bc$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a*b - 3*a*c + 4*b*c
Exercise 19, problem 7, p. 50
$2x^{3} - x^{2} - 5x + 3$ from $3x^{3} + 2x^{2} - 3x - 5$.
Printed answer:
- $x^{3} + 3x^{2} + 2x - 8$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**3 + 3*x**2 + 2*x - 8
Exercise 19, problem 8, p. 50
$7x^{2} - 5x + 1 - a$ from $x^{3} - x + 1 - a$.
Printed answer:
- $x^{3} - 7x^{2} + 4x$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**3 - 7*x**2 + 4*x
Exercise 19, problem 9, p. 50
$7b^{3} + 8c^{3} - 15abc$ from $9b^{3} + 3abc - 7c^{3}$.
Printed answer:
- $2b^{3} + 18abc - 15c^{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*b**3 + 18*a*b*c - 15*c**3

Exercise 20

Exercise 20, problem 1, p. 52
$a - b - (b - c) - a + 2b$.
Printed answer:
- $c$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes c
Exercise 20, problem 10, p. 52
$(a - b + c) - (b - a - c) + (a + b - 2c)$.
Printed answer:
- $3a - b$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*a - b
Exercise 20, problem 11, p. 52
$3x - [-2y - (2y - 3x) + z] + [x - (y - 2z - x)]$.
Printed answer:
- $2x + 3y + z$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*x + 3*y + z
Exercise 20, problem 12, p. 52
$x - [2x + (x - 2y) + 2y] - 3x - \bigl\{4x - [(x + 2y) - y]\bigr\}$.
Printed answer:
- $-8x + y$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -8*x + y
Exercise 20, problem 13, p. 52
$x - [y + z - x - (x + y) - z] + (3 x - \Vinc{2y + z})$.
Printed answer:
- $6z - 2y-z$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem 6*z - 2*y - z
Exercise 20, problem 14, p. 52
$ax + by + cz - ay + az - bx = (a - b)x - (a - b)y + (a + c)z$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- identity: no printed answer to check
Exercise 20, problem 15, p. 52
$ax + az + by - cz - ay + cx$.
Printed answer:
- $(a + c)x - (a - b)y + (a - c)z$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (a + c)*x - (a - b)*y + (a - c)*z
Exercise 20, problem 16, p. 52
$2ax - 3ay - 4by + 5cx - 6bz - 7cz$.
Printed answer:
- $(2a + 5c)x - (3a + 4b)y - (6b + 7c)z$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (2*a + 5*c)*x - (3*a + 4*b)*y - (6*b + 7*c)*z
Exercise 20, problem 17, p. 52
$az - bmy + 3 cz - anx - cny + acx$.
Printed answer:
- $(ac - an)x - (bm + cn)y + (a + 3c)z$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (a*c - a*n)*x - (b*m + c*n)*y + (a + 3*c)*z
Exercise 20, problem 18, p. 52
$mnx - x - mny - y + mnz + z$.
Printed answer:
- $(mn - 1)x - (mn + 1)y + (mn + 1)z$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (m*n - 1)*x - (m*n + 1)*y + (m*n + 1)*z
Exercise 20, problem 2, p. 52
$x - [x - (a - b) + a - y]$.
Printed answer:
- $y - b$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes y - b
Exercise 20, problem 3, p. 52
$3x - \bigl\{2y - [-7c - 2x] + y\bigr\}$.
Printed answer:
- $x - 3y - 7c$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x - 3*y - 7*c
Exercise 20, problem 4, p. 52
$5a - [7 - (2b + 5) - 2a]$.
Printed answer:
- $7a + 2b-2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 7*a + 2*b - 2
Exercise 20, problem 5, p. 52
$x - [2x + (3a - 2x) - 5a]$.
Printed answer:
- $2a + x$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a + x
Exercise 20, problem 6, p. 52
$x - [15y - (13z + 12x)]$.
Printed answer:
- $13x - 15y + 13z$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 13*x - 15*y + 13*z
Exercise 20, problem 7, p. 52
$2a - b + [4c - (b + 2c)]$.
Printed answer:
- $2a - 2b + 2c$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a - 2*b + 2*c
Exercise 20, problem 8, p. 52
$5a - \bigl\{b + [3c - (2b - c)]\bigr\}$.
Printed answer:
- $5a + b - 4c$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 5*a + b - 4*c
Exercise 20, problem 9, p. 52
$7x - \bigl\{5y - [3z - (3x + z)]\bigr\}$.
Printed answer:
- $4x - 5y + 2z$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*x - 5*y + 2*z