Question

Find the solutions for the following.
(i)Subtract \[4a - 7ab + 3b + 12\] from $12a - 9ab + 5b - 3$
(ii)Subtract $3xy + 5yz - 7zx$ from $5xy - 2yz - 2zx + 10xyz$
(iii)Subtract $4{p^2}q - 3pq + 5p{q^2} - 8p + 7q - 10$ from $18 - 3p - 11q + 5pq - 2p{q^2} + 5{p^2}q$

Answer 1

Hint: Let us divide similar degree terms from both the equations and place them together for further subtraction. To subtract one equation from another we have to subtract the similar terms from both the expressions

Complete step-by-step solution:
(i) Subtract $4a - 7ab + 3b + 12$ from $12a - 9ab + 5b - 3$
In equation format is, $\left( {12a - 9ab + 5b - 3} \right) - \left( {4a - 7ab + 3b + 12} \right)$ $ \to eq(1)$
As we see, both the equations are polynomials with two variables, ‘a’ and ‘b’ and both the equations are of degree 2.
We need to group the similar terms together to further solve the subtraction.
\[\left( {12a,4a} \right);\left( {9ab,7ab} \right);\left( {5b,3b} \right);\left( {3,12} \right)\] These are the similar terms from the equations.
In the eq(1) send the subtraction(-) into the second equation.
After doing the above step, eq(1) will be as follows
$ \to 12a - 9ab + 5b - 3 - 4a + 7ab - 3b - 12$
Group the similar terms and then subtract
$
   \to \left( {12a - 4a} \right) + \left( {7ab - 9ab} \right) + \left( {5b - 3b} \right) + \left( { - 3 - 12} \right) \\
   \to 8a + \left( { - 2ab} \right) + 2b + \left( { - 15} \right) \\
   \to 8a - 2ab + 2b - 15 \\
$
$\therefore \left( {12a - 9ab + 5b - 3} \right) - \left( {4a - 7ab + 3b + 12} \right)$ is $8a - 2ab + 2b - 15$
(ii) Subtract $3xy + 5yz - 7zx$ from $5xy - 2yz - 2zx + 10xyz$
In equation format is, $\left( {5xy - 2yz - 2zx + 10xyz} \right) - \left( {3xy + 5yz - 7zx} \right)$ $ \to eq(1)$
As we see, both the equations are polynomials with three variables, ‘x’, ‘y’ and ‘z’ and the first equation is of degree 3 and the second equation is of degree 2. So the resultant equation will have the degree 3 as 3 is greater than 2.
We need to group the similar terms together to further solve the subtraction.
$\left( {3xy,5xy} \right);\left( {5yz,2yz} \right);\left( {7zx,2zx} \right);\left( {10xyz} \right)$ These are the similar terms from the equations.
In the eq(1) send the subtraction(-) into the second equation.
After doing the above step, eq(1) will be as follows
$ \to 5xy - 2yz - 2zx + 10xyz - 3xy - 5yz + 7zx$
Group the similar terms and then subtract
$
   \to \left( {5xy - 3xy} \right) + \left( { - 2yz - 5yz} \right) + \left( {7zx - 2zx} \right) + 10xyz \\
   \to 2xy + \left( { - 7yz} \right) + 5zx + 10xyz \\
   \to 2xy - 7yz + 5zx + 10xyz \\
$
$\therefore \left( {5xy - 2yz - 2zx + 10xyz} \right) - \left( {3xy + 5yz - 7zx} \right)$ is \[2xy - 7yz + 5zx + 10xyz\]
(iii) Subtract $4{p^2}q - 3pq + 5p{q^2} - 8p + 7q - 10$ from $18 - 3p - 11q + 5pq - 2p{q^2} + 5{p^2}q$
In equation format is, $\left( {18 - 3p - 11q + 5pq - 2p{q^2} + 5{p^2}q} \right) - \left( {4{p^2}q - 3pq + 5p{q^2} - 8p + 7q - 10} \right)$ $ \to eq(1)$
As we see, both the equations are polynomials with two variables, ‘p’ and ‘q’ and both the equations are of degree 3.
We need to group the similar terms together to further solve the subtraction.
$\left( {4{p^2}q,5{p^2}q} \right);\left( {3pq,5pq} \right);\left( {5p{q^2},2p{q^2}} \right);\left( {8p,3p} \right);\left( {7q,11q} \right);\left( {10,18} \right)$ These are the similar terms from the equations.
In the eq(1) send the subtraction(-) into the second equation.
After doing the above step, eq(1) will be as follows
$ \to 18 - 3p - 11q + 5pq - 2p{q^2} + 5{p^2}q - 4{p^2}q + 3pq - 5p{q^{^2}} + 8p - 7q + 10$
Group the similar terms and then subtract
$
   \to \left( {18 + 10} \right) + \left( {8p - 3p} \right) + \left( { - 11q - 7q} \right) + \left( {5pq + 3pq} \right) + \left( { - 2p{q^2} - 5p{q^2}} \right) + \left( {5{p^2}q - 4{p^2}q} \right) \\
   \to 28 + 5p + \left( { - 18q} \right) + 8pq + \left( { - 7p{q^2}} \right) + {p^2}q \\
   \to 28 + 5p - 18q + 8pq - 7p{q^2} + {p^2}q \\
$
Arrange the terms in an order
$\therefore $ ${p^2}q - 7p{q^2} + 8pq + 5p - 18q + 28$
$\therefore \left( {18 - 3p - 11q + 5pq - 2p{q^2} + 5{p^2}q} \right) - \left( {4{p^2}q - 3pq + 5p{q^2} - 8p + 7q - 0} \right)$ is ${p^2}q - 7p{q^2} + 8pq + 5p - 18q + 28$

Note:The degree of an equation is the highest degree of its terms while the degree of the terms in the equation is the sum of the exponents of the variables in it.