SUBTRACTION
Excerpts
SUBTRACTION
The principle is clear; namely, ***The subtraction of any number gives the same result as the addition of that number with the opposite sign.***
SUBTRACTION
*To subtract one number from another, consider the sign of the subtrahend changed and add.*
SUBTRACTION
The parenthesis indicates that the numbers enclosed are considered as one number.
SUBTRACTION
***Any number of terms may be removed from a parenthesis preceded by the minus sign by changing the sign of each term*.**
SUBTRACTION
***Any number of terms may be enclosed in a parenthesis preceded by the minus sign by changing the sign of each term*.**
Equations
SUBTRACTION
a + (b - c - x) = a + b - c - xRemoving the parenthesis preceded by a plus sign leaves the terms inside it unchanged, so a plus-preceded parenthesis can simply be dropped.
SUBTRACTION
a + c - d + e = a + (c - d + e)Terms may be enclosed in a parenthesis preceded by a plus sign without changing their signs, which is the converse of removing such a parenthesis.
SUBTRACTION
x - (y + z - c) = x - y - z + cRemoving a parenthesis preceded by a minus sign changes the sign of each term inside it, so subtracting the bracketed sum gives the same result as subtracting each term.
SUBTRACTION
a - b - c + d = a - (b + c - d)Enclosing terms in a parenthesis preceded by a minus sign requires changing the sign of each enclosed term, which is the converse of removing such a parenthesis.
Problems
Exercise 16
Exercise 16, problem 1
From $5a^{3}$ take $3a^{3}$.
Printed answer:- $2a^3$.
verified: the printed answer passed a computed check
How it was checked
identity: passes2*a**3
Exercise 16, problem 10
From $9x - 4y + 3z$ take $5x - 3y + z$.
Printed answer:- $4x - y + 2z$.
verified: the printed answer passed a computed check
How it was checked
identity: passes4*x - y + 2*z
Exercise 16, problem 11
Subtract $3x^{4} - x^{2} + 7x - 14$ from $11x^{4} - 2x^{3} - 8x$.
Printed answer:- $8x^4 - 2x^3 + 4x^2 - 15x + 14$.
unverified: no computed check settled this one (yet)
How it was checked
identity: FLAG-INTERPRETED8*x**4 - 2*x**3 + 4*x**2 - 15*x + 14
Exercise 16, problem 12
From $10a^{2}b^{2} + 15ab^{2} + 8a^{2}b$ take $-10a^{2}b^{2} + 15ab^{2} - 8a^{2}b$.
Printed answer:- $20a^2b^2 + 16a^2b$.
verified: the printed answer passed a computed check
How it was checked
identity: passes20*a**2*b**2 + 16*a**2*b
Exercise 16, problem 13
Subtract $1- x + x^{2} - 3x^{3}$ from $x^{3} - 1 + x^{2} - x$.
Printed answer:- $4x^3 - 2$.
verified: the printed answer passed a computed check
How it was checked
identity: passes4*x**3 - 2
Exercise 16, problem 14
From $x^{m} - 2x^{2m} + x^{3m}$ take $2x^{3m} - x^{2m} - x^{m}$.
Printed answer:- $2x^m - x^{2m} - x^{3m}$
verified: the printed answer passed a computed check
How it was checked
identity: passes2*x**m - x**(2*m) - x**(3*m)
Exercise 16, problem 15
Subtract $a^{2n} + a^{n}x^{n} + x^{2n}$ from $3a^{2n} - 17a^{n}x^{n} - 8x^{2n}$.
Printed answer:- $2a^{2n} - 18a^nx^n - 9x^{2n}$.
verified: the printed answer passed a computed check
How it was checked
identity: passes2*a**(2*n) - 18*a**n*x**n - 9*x**(2*n)
Exercise 16, problem 16
From $\frac{2}{3}a^{2} - \frac{5}{2}a - 1$ take $-\frac{2}{3}a^{2} + a -\frac{1}{2}$.
Printed answer:- $\frac{4}{3}a^2 - \frac{7}{2}a - \frac{1}{2}$.
verified: the printed answer passed a computed check
How it was checked
identity: passesRational(4,3)*a**2 - Rational(7,2)*a - Rational(1,2)
Exercise 16, problem 17
From $x^{5} + 3xy^{4}$ take $x^{5} + 2x^{4}y + 3x^{3}y^{2} - 2xy^{4} + y^{5}$.
Printed answer:- $-2x^4y - 3x^3y^2 + 5xy^4 - y^5$.
verified: the printed answer passed a computed check
How it was checked
identity: passes-2*x**4*y - 3*x**3*y**2 + 5*x*y**4 - y**5
Exercise 16, problem 18
From $x$ take $y - a$.
Printed answer:- $x - y + a$.
verified: the printed answer passed a computed check
How it was checked
identity: passesx - y + a
Exercise 16, problem 19
From $6a^3 + 4a + 7$ take the sum of $2a^3 + 4a^2 + 9$ and $4a^3 - a^2 + 4a - 2$.
Printed answer:- $-3a^2$.
verified: the printed answer passed a computed check
How it was checked
identity: passes-3*a**2
Exercise 16, problem 2
From $7a^{2}b$ take $-5a^{2}b$.
Printed answer:- $12a^2b$.
verified: the printed answer passed a computed check
How it was checked
identity: passes12*a**2*b
Exercise 16, problem 20
Subtract $3x - 7x^3 + 5x^2$ from the sum of $2 + 8x^2 - x^3$ and $2x^3 - 3x^2 + x - 2$.
Printed answer:- $8x^3 - 2x$.
verified: the printed answer passed a computed check
How it was checked
identity: passes8*x**3 - 2*x
Exercise 16, problem 21
What must be subtracted from $15y^3 + z^3 + 4yz^2 - 5z^2x - 2xy^2$ to leave a remainder of $6x^3 - 12y^3 + 4z^3 - 2xy^2 + 6z^2x$?
Printed answer:- $27y^3 - 3z^3 - 6x^3 + 4yz^2 - 11z^2x$.
verified: the printed answer passed a computed check
How it was checked
solve: passes, with the problem read into an equation27*y**3 - 3*z**3 - 6*x**3 + 4*y*z**2 - 11*z**2*x
Exercise 16, problem 22
How much must be added to $x^3 - 4x^2 + 16x$ to produce $x^3 + 64$?
Printed answer:- $4x^2 - 16x + 64$.
verified: the printed answer passed a computed check
How it was checked
solve: passes, with the problem read into an equation4*x**2 - 16*x + 64
Exercise 16, problem 23
To what must $4a^2 - 6b^2 + 8bc - 6ab$ be added to produce zero?
Printed answer:- $-4a^2 + 6b^2 - 8bc + 6ab$.
verified: the printed answer passed a computed check
How it was checked
solve: passes, with the problem read into an equation-4*a**2 + 6*b**2 - 8*b*c + 6*a*b
Exercise 16, problem 24
From what must $2x^4 - 3x^2 + 2x - 5$ be subtracted to produce unity?
Printed answer:- $2x^4 - 3x^2 + 2x - 4$.
verified: the printed answer passed a computed check
How it was checked
solve: passes, with the problem read into an equation2*x**4 - 3*x**2 + 2*x - 4
Exercise 16, problem 25
What must be subtracted from the sum of $3a^3 + 7a + 1$ and $2a^2 - 5a - 3$ to leave a remainder of $2a^2 - 2a^3 - 4$?
Printed answer:- $5a^3 + 2a + 2$.
verified: the printed answer passed a computed check
How it was checked
solve: passes, with the problem read into an equation5*a**3 + 2*a + 2
Exercise 16, problem 26
From the difference between $10a^{2}b + 8ab^{2} - 8a^{2}b^{2} - b^{3}$ and $5a^{2}b - 6ab^{3} - 7a^{2}b^{2}$ take the sum of $10a^{2}b^{2} + 15ab^{2} + 8a^{2}b$ and $8a^{2}b - 5ab^{2} + a^{2}b^{2}$.
Printed answer:- $-11a^2b + 4ab^2 - 12a^2b^2 - b^3$.
unverified: no computed check settled this one (yet)
How it was checked
identity: the record may be misread-11*a**2*b + 4*a*b**2 - 12*a**2*b**2 - b**3
Exercise 16, problem 27
What must be added to $a$ to make $b$?
Printed answer:- $b - a$.
verified: the printed answer passed a computed check
How it was checked
solve: passes, with the problem read into an equationb - a
Exercise 16, problem 28
By how much does $3x - 2$ exceed $2x + 1$?
Printed answer:- $x - 3$.
verified: the printed answer passed a computed check
How it was checked
identity: passesx - 3
Exercise 16, problem 29
In $y$ years a man will be 40 years old. What is his present age?
Printed answer:- $40 - y$ yrs.
verified: the printed answer passed a computed check
How it was checked
solve: passes, with the problem read into an equation40 - y
Exercise 16, problem 3
Subtract $7xy^{3}$ from $-2xy^{3}$.
Printed answer:- $-9xy^3$.
verified: the printed answer passed a computed check
How it was checked
identity: passes-9*x*y**3
Exercise 16, problem 30
How many hours will it take to go 23 miles at $a$ miles an hour?
Printed answer:- $\frac{23}{a}$ hrs.
verified: the printed answer passed a computed check
How it was checked
solve: passes, with the problem read into an equation23/a
Exercise 16, problem 4
From $-3x^{m}y$ take $-7x^{m}y$.
Printed answer:- $4x^my$.
verified: the printed answer passed a computed check
How it was checked
identity: passes4*x**m*y
Exercise 16, problem 5
Subtract $3ax$ from $8x^{2}$.
Printed answer:- $8x^2 - 3ax$.
verified: the printed answer passed a computed check
How it was checked
identity: passes8*x**2 - 3*a*x
Exercise 16, problem 6
From $5xy$ take $-7by$.
Printed answer:- $5xy + 7by$.
verified: the printed answer passed a computed check
How it was checked
identity: passes5*x*y + 7*b*y
Exercise 16, problem 7
What is the difference between $4a^{m}$ and $2a^{m}$?
Printed answer:- $2a^m$.
verified: the printed answer passed a computed check
How it was checked
identity: passes2*a**m
Exercise 16, problem 8
From the difference between $5a^{2}x$ and $-3a^{2}x$ take the sum of $2a^{2}x$ and $-3a^{2}x$.
Printed answer:- $9 ax$.
unverified: no computed check settled this one (yet)
How it was checked
identity: FLAG-INTERPRETED9*a*x
Exercise 16, problem 9
From $2a + b + 7c$ take $5a + 2b - 7c$.
Printed answer:- $-3a - b + 14c$.
verified: the printed answer passed a computed check
How it was checked
identity: passes-3*a - b + 14*c
Exercise 17
Exercise 17, problem 1
$x + (a + b) + y + (c - d) + (x - y)$.
Printed answer:- $2x + a + b + c - d$.
verified: the printed answer passed a computed check
How it was checked
identity: passes2*x + a + b + c - d
Exercise 17, problem 10
$2 a^2 + 3a^3x-ab^2 + by^2$.
Printed answer:- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handles
Exercise 17, problem 11
$10 m^3 +31m^2-20m-21$.
Printed answer:- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handles
Exercise 17, problem 12
$ax^4 - x^3 + 2x - 2ax^2$.
Printed answer:- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handles
Exercise 17, problem 13
$a^4 + a^3x + a^2x^2 - ax^3- 4x^4$.
Printed answer:- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handles
Exercise 17, problem 14
$a^4 + a^3-6a^2+a + 3$.
Printed answer:- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handles
Exercise 17, problem 15
$6a^3 - 17 a^2x + 14 ax^2 - 3x^3$.
Printed answer:- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handles
Exercise 17, problem 16
$ax^3 + 2ax^2 + ax + 2a$.
Printed answer:- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handles
Exercise 17, problem 17
A man pumps $x$ gallons of water into a tank each day, and draws off $y$ gallons each day. How much water will remain in the tank at the end of five days?
Printed answer:- $5(x - y)$.
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handles5*(x - y)
Exercise 17, problem 18
Two men are 150 miles apart, and approach each other, one at the rate of $x$ miles an hour, the other at the rate of $y$ miles an hour. How far apart will they be at the end of seven hours?
Printed answer:- $150 - 7(x + y)$.
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handles150 - 7*(x + y)
Exercise 17, problem 19
Eight years ago A was $x$ years old. How old is he now?
Printed answer:- $x + 8$ yrs.
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handlesx + 8
Exercise 17, problem 2
$a + (b - c) - b + (a + c) + (c - a)$.
Printed answer:- $a + c$.
verified: the printed answer passed a computed check
How it was checked
identity: passesa + c
Exercise 17, problem 20
A had $x$ dollars, but after giving $35 to B he has one-third as much as B. How much has B?
Printed answer:- $3(x - 35)$ dols.
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handles3*(x - 35)
Exercise 17, problem 3
$a^{2}b - (a^{3} + b^{3}) - a^{3} - (ab^{2} - a^{2}b) -(b^{3} - a^{3})$.
Printed answer:- $2a^2b-a^3-2b^3-ab^2$.
verified: the printed answer passed a computed check
How it was checked
identity: passes2*a**2*b - a**3 - 2*b**3 - a*b**2
Exercise 17, problem 4
$xy - (x^{2} + y^{2}) - y^{2} - (x^{2} - 2xy) - (y^{2} - x^{2})$.
Printed answer:- $3xy - x^2 - 3y^2$.
verified: the printed answer passed a computed check
How it was checked
identity: passes3*x*y - x**2 - 3*y**2
Exercise 17, problem 5
$(a + b - c) - (a - b + c) + (b - a - c) - (c - a - b)$.
Printed answer:- $4b - 4c$.
verified: the printed answer passed a computed check
How it was checked
identity: passes4*b - 4*c
Exercise 17, problem 6
$(x - y + z) + (x + y + z) - (y + x + z) - (z + x + y)$.
Printed answer:- $-2y$.
verified: the printed answer passed a computed check
How it was checked
identity: passes-2*y
Exercise 17, problem 7
$a - (3b - 2c + a) - (2b - a - c) - (6 - c + a)$.
Printed answer:- $-6b + 4c$.
unverified: no computed check settled this one (yet)
How it was checked
identity: the printed answer does not match the problem-6*b + 4*c
Exercise 17, problem 8
$\frac{1}{2}a - \frac{1}{2}c - (\frac{2}{3}b - \frac{1}{2}c) - (a + \frac{1}{4}c - \frac{1}{3}b) - (\frac{2}{3}b - \frac{1}{4}c - \frac{1}{2}a)$.
Printed answer:- $-b$.
verified: the printed answer passed a computed check
How it was checked
identity: passes-b
Exercise 17, problem 9
$x - y + 2c - d$.
Printed answer:- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handles