Download Transforming Equations: Multiplication and Division

Transforming Equations:
Multiplication and Division
September 22, 2011
Transforming Equations
Objective To solve equations using
multiplication or division.
Transforming Equations
At a hardware store, small construction
supplies are often sold by the pound rather
than by the number of items.
Suppose a pound of roof nails costs the same
as a pound of floor nails. You would expect to
pay the same price for two pounds of roof nails
as for two pounds of floor nails, and the same
price for one-half pound of roof nails as for
one-half pound of floor nails.
Multiplication Property of Equality
If a, b, and c are any real numbers, and
𝑎 = 𝑏, then
𝒂𝒄 = 𝒃𝒄 and 𝒄𝒂 = 𝒄𝒃
If equal numbers are multiplied by the same
number, the products are equal.
Division Property of Equality
If a and b are any real numbers , c is any
nonzero real number, and 𝑎 = 𝑏, then
𝒂 𝒃
=
𝒄 𝒄
If equal numbers are divided by the same
nonzero number, the quotients are equal.
Transforming an Equation into
an Equivalent Equation
These properties give you two more ways to
transform an equation into an
equivalent
equation.
Transforming an Equation into
an Equivalent Equation
Transformation by Multiplication:
Multiply each side of a given equation by
the same nonzero real number.
Transformation by Division:
Divide each side of a given equation by
the same nonzero real number.
Example 1
Solve
6𝑥 = 222.
Solution
6𝑥 = 222
Copy the equation.
6𝑥 222
=
6
6
Divide to each side by 6.
𝑥 = 37
Simplify.
Example 1
Solve
6𝑥 = 222.
Check
6𝑥 = 222
Copy the equation.
6 37 = 222
Substitute 37 for x.
222 = 222
 The solution set is {37}.
Example 2
Solve
8=
2
− 𝑡.
3
Solution
2
8=− 𝑡
3
3
3
2
− 8 =− − 𝑡
2
2
3
−12 = 𝑡
Copy the equation.
To get t alone on one
side, multiply each
3
side by − , the
2
reciprocal of −
Simplify.
2
3
Example 2
Solve
8=
2
− 𝑡.
3
Check
2
8=− 𝑡
3
Copy the equation.
2
8 = − −12
3
Substitute 12 for t.
8=8
 The solution set is {12}.
Equivalent Equations
You know that zero cannot be a divisor. Do
you know why zero is not allowed as a
multiplier in transforming an equation?
Look at the following equations.
1.
2.
3.
4.
5𝑧 = 45
0 ∙ 5𝑧 = 0 ∙ 45
0 ∙ 5 𝑧 = 0 ∙ 45
0∙𝑧 =0
Equivalent Equations
Equation (1) had just one root, namely 9.
Equation (4) is satisfied by any real number.
Since they do not have the same solution set,
Equations (1) and (4)
are not equivalent.
When transforming an equation,
never multiply by zero!
Class work
Oral Exercises
p 104: 1-18
Homework
p 104: 3-39 mult of 3,
p 105: prob 7-15 odd,
p 106 MR