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LESSON
2.4 Variables and Equations
Biology Lotus flowers like the one
shown can be grown from seeds
hundreds of years old. You will see
how an equation can be used to
estimate the year when an ancient
lotus seed was formed.
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LESSON
2.4 Variables and Equations
An equation is a mathematical sentence formed by placing an equal
sign, =, between two expressions. A solution of an equation with a
variable is a number that produces a true statement when it is
substituted for the variable.
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LESSON
2.4 Variables and Equations
EXAMPLE
1
Writing Verbal Sentences as Equations
Verbal Sentence
Equation
The sum of x and 6 is 9.
x+6=9
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LESSON
2.4 Variables and Equations
EXAMPLE
1
Writing Verbal Sentences as Equations
Verbal Sentence
Equation
The sum of x and 6 is 9.
x+6=9
The difference of 12 and y is 15.
12 – y = 15
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LESSON
2.4 Variables and Equations
EXAMPLE
1
Writing Verbal Sentences as Equations
Verbal Sentence
Equation
The sum of x and 6 is 9.
x+6=9
The difference of 12 and y is 15.
12 – y = 15
The product of –4 and p is 32.
–4p = 32
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LESSON
2.4 Variables and Equations
EXAMPLE
1
Writing Verbal Sentences as Equations
Verbal Sentence
Equation
The sum of x and 6 is 9.
x+6=9
The difference of 12 and y is 15.
12 – y = 15
The product of –4 and p is 32.
–4p = 32
The quotient of n and 2 is 9.
n
=9
2
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LESSON
2.4 Variables and Equations
EXAMPLE
2
Checking Possible Solutions
Tell whether 9 or 7 is a solution of x – 5 = 2.
Substitute 9 for x.
x–5=2
?
9–5 =2
4=2
ANSWER
9 is not a solution.
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LESSON
2.4 Variables and Equations
EXAMPLE
2
Checking Possible Solutions
Tell whether 9 or 7 is a solution of x – 5 = 2.
Substitute 9 for x.
x–5=2
?
9–5 =2
Substitute 7 for x.
x–5=2
7–5 =2
?
4=2
ANSWER
9 is not a solution.
2=2
ANSWER
7 is a solution.
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LESSON
2.4 Variables and Equations
Solving Equations Finding all solutions of an
equation is called solving the equation. You can use
mental math to solve a simple equation by thinking of
the equation as a question.
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LESSON
2.4 Variables and Equations
EXAMPLE
3
Solving Equations Using Mental Math
Equation
Question
x + 3 = 11
What number
plus 3 equals 11?
Solution
8
Check
8 + 3 = 11
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LESSON
2.4 Variables and Equations
EXAMPLE
3
Solving Equations Using Mental Math
Equation
Question
Solution
Check
x + 3 = 11
What number
plus 3 equals 11?
8
8 + 3 = 11
16 – m = 9
16 minus what
number equals 9?
7
16 – 7 = 9
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LESSON
2.4 Variables and Equations
EXAMPLE
3
Solving Equations Using Mental Math
Equation
Question
Solution
Check
x + 3 = 11
What number
plus 3 equals 11?
8
8 + 3 = 11
16 – m = 9
16 minus what
number equals 9?
7
16 – 7 = 9
20 = 5t
20 equals 5 times
what number?
4
20 = 5(4)
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LESSON
2.4 Variables and Equations
EXAMPLE
3
Solving Equations Using Mental Math
Equation
Question
Solution
Check
x + 3 = 11
What number
plus 3 equals 11?
8
8 + 3 = 11
16 – m = 9
16 minus what
number equals 9?
7
16 – 7 = 9
20 = 5t
20 equals 5 times
what number?
4
20 = 5(4)
y
= –3
6
What number divided
by 6 equals –3?
–18
–18
= –3
6
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LESSON
2.4 Variables and Equations
EXAMPLE
4
Writing and Solving an Equation
From 1998 to 2002, biologist Jane Shen-Miller grew
several lotus plants from ancient seeds she found in
China. The oldest seed was about 500 years old. Estimate
the year when this seed was formed.
SOLUTION
First write a verbal model for this situation.
Year seed
was formed
+
Age of seed
when it sprouted
=
Year seed
sprouted
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LESSON
2.4 Variables and Equations
EXAMPLE
4
Writing and Solving an Equation
From 1998 to 2002, biologist Jane Shen-Miller grew
several lotus plants from ancient seeds she found in
China. The oldest seed was about 500 years old.
Estimate the year when this seed was formed.
SOLUTION
Let x represent the year when the seed was formed. Because you are only trying
to estimate x (rather than determine x precisely), you can use 2000 for the year
when the seed sprouted. This year simplifies mental calculations and lies within
the given time period, 1998–2002.
x + 500 = 2000
1500 + 500 = 2000
ANSWER
Substitute for quantities in verbal model.
Use mental math to solve for x.
Because x = 1500, the seed was formed around the year 1500.
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