ADD SUBTRACT AND MULTIPLY LINEAR EXPRESSIONS
Adding and subtracting linear expressions is the same as the procedure used in combining like terms. When adding polynomials, simply drop the parenthesis and combine like terms. When subtracting polynomials, distribute the negative first, then combine like terms.
When two linear expressions are multiplied, multiply each term in one linear expression by each term in other linear expression.
Example 1 :
Add (6a + 3) and (4a - 2).
Solution :
= (6a + 3) + (4a - 2)
= 6a + 3 + 4a - 2
= 10a + 1
Example 2 :
Add (5y + 8 + 3z) and (4y - 5).
Solution :
= (5y + 8 + 3z) + (4y - 5)
= 5y + 8 + 3z + 4y - 5
= 9y + 3z + 3
Example 3 :
Subtract (6a - 3b) from (-8a + 9b).
Solution :
= (-8a + 9b) - (6a - 3b)
= -8a + 9b - 6a + 3b
= -14a + 12b
Example 4 :
Subtract (2x + 3y - 5z) from (5x - 4y - 5z).
Solution :
= (5x - 4y - 5z) - (2x + 3y - 5z)
= 5x - 4y - 5z - 2x - 3y + 5z
= 3x - 7y
Example 5 :
Multiply (3x - 7) and (7x - 3).
Solution :
= (3x - 7)(7x - 3)
= 21x2 - 9x - 49x + 21
= 21x2 - 58x + 21
Example 6 :
Multiply (3a - 2b) and (2a + 3b).
Solution :
= (3a - 2b)(2a + 3b)
= 6a2 + 9ab - 4ab - 6b2
= 6a2 + 9ab - 4ab - 6b2
= 6a2 + 5ab - 6b2
Example 7 :
Multiply (p + q + r) and (p + q - r).
Solution :
= (p + q + r)(p + q - r)
Let x = p + q.
= (x + r)(x - r)
Using Algebraic Identity a2 - b2 = (a + b)(a - b),
= x2 - r2
Substitute x = p + q.
= (p + q)2 - r2
Using Algebraic Identity (a + b)2 = a2 + 2ab + b2.
= p2 + 2pq + q2 - r2
Dividing Linear Expressions
When a linear expression is divided by another linear expression, factor the expressions in both numerator and denominator and cancel out the common factor.
Example 8 :
Divide (5x + 20) by (5x + 35).
Solution :
= (5x + 20)/(5x + 35)
= [5(x + 4)]/[5(x + 7)]
= (x + 4)/(x + 7)
Example 9 :
Divide (5x - 25) by (3x - 15).
Solution :
= (5x - 25)/(3x - 15)
= 5(x - 5)/3(x - 5)
= 5/3
Example 10 :
Divide (2 - x) by (4x - 8).
Solution :
= (2 - x)/(4x - 8)
= (2 - x)/4(x - 2)
= -(x - 2)/4(x - 2)
= -1/4
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