Download Multiplication and Division Properties of Equality

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Sect 3.3 - Multiplication and Division Properties of
Equality
Objective 1:
Multiplication and Division Properties of Equality.
Multiplication and division properties of equality work in a similar fashion to
the addition and subtraction properties of equality. Consider the
following:
Solve the following:
Ex. 1
3x = 27
Ex. 2
w
5
= 11
Solution:
Solution:
Since 3•9 = 27, then x = 9.
Since 55 ÷ 5 = 11, then
Or, 27 ÷ 3 = 9.
€ w = 55. Or, 11•5 = 55.
In example #1, the second way we solved the problem was by dividing 27
by 3. On the left side, if we divide 3x by 3, we get x. So, by dividing both
sides of 3x = 27, we get the equation x = 9. In example #2, the second way
we solved the problem was by multiplying 1.1 by 5. On the left side. if we
multiply
w
5
by 5, we get w. So, by multiplying both sides of
w
5
= 11 by 5,
we get w = 55. This means we can multiply or divide both sides of an
equation by any non-zero quantity without changing the answer. These are
known as the Multiplication and Division Properties of Equality.
€
€
Multiplication and Division Properties of Equality:
If a, b, and c are algebraic expressions with c ≠ 0, and if a = b, then
1.
ac = bc is equivalent to a = b.
Multiplication Property
2.
a
c
=
b
c
is equivalent to a = b.
Division Property
There are several comments that need to be made about these properties.
First, we cannot multiply both sides of an equation by zero. If we do, we will
get an equation that has a different solution than the original equation and
hence it will not be an equivalent equation. We also cannot divide both
sides by zero since division by zero is undefined.
Solve the following:
Ex. 3
– 25x = – 475
Ex. 4
– 17x = 51
€
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Solution:
Divide both sides by – 25
to solve for x:
– 25x = – 475
– 25
– 25
x = 19
Check:
– 25(19) = – 475
– 475 = – 475 True
So, x = 19.
x
−11
Ex. 5
Solution:
Divide both sides by – 17
to solve for x:
– 17x = 51
– 17
– 17
x=–3
Check:
– 17(– 3) = 51
51 = 51 True
So, x = – 3.
=–4
Solution:
Multiply both sides by – 11
€ reciprocal of
x
( −11
) = – 11(– 4)
3
44
Check:
€
€
Ex. 8
– x = – 65
Solution:
Divide by – 1:
−x
−65
=
Check:
€
−1
(– 15) = – 5
– 5 = – 5 True
– 32p = 0
Solution:
Divide by – 32:
−32p
0
=
−32
−32
p=0
– (65) = – 65
– 65 = – 65 True
So, x = 65.
Objective 2:
1
3
So, t = – 15.
x = 65
€
to solve for t:
t = – 15
Check:
=–4
−11
€
–4=–4
So, x = 44.
−1
4
7
( 31 t) = 3(– 5)
x = 44
Ex.€7
=–5
Solution:
Multiply both sides by the
€ to solve for x:
– 11
1
t
3
Ex. 6
€
Check:
€
– 32(0) = 0
0 = 0 True
So, p = 0.
Comparing the properties of equality
In solving linear equations, it is important to identify which property needs
to be used to solve an equation. We do this by examining the operation
between the variable and the number next to it. Whatever that operation is,
we use the reverse process to solve the equation.
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Solve, but do not check the answer:
Ex. 9
– 7x = 42
Solution:
Since the operation between
– 7 and x is multiplication, we
need to divide both sides by – 7:
– 7x = 42
–7 –7
x=–6
Ex. 11
–
x
7
= 42
Solution:
Since the operation between
– 7 and x is division, we need
€to multiply both sides by – 7:
(
–7 –
x
7
) = – 7(42)
x = – 294
€
Ex. 13
4(y – 3) + 6 = 7 – 13
Solution:
Distribute the 4:
4y – 12 + 6 = 7 – 13
Combine like terms:
4y – 6 = – 6
Add 6 to both sides:
4y – 6 = – 6
+6=+6
4y
=0
Divide both sides by 4:
4y = 0
4
4
y=0
Ex. 10
– 7 + x = 42
Solution:
Since the operation between
– 7 and x is addition, we
need to add 7 to both sides:
– 7 + x = 42
+7
=+7
x = 49
Ex. 12
– 7 – x = 42
Solution:
Since the operation between
– 7 and x is subtraction, we
need to add 7 to both sides:
– 7 – x = 42
+7
=+7
– x = 49
To solve for x, divide both
sides by – 1
– x = 49
–1
–1
x = – 49
Ex. 14
7y – 3(2y + 3) = – 9
Solution:
Distribute the – 3:
7y – 6y – 9 = – 9
Combine like terms:
y–9=–9
Add 9 to both sides:
y–9=–9
+9=+9
y=0