Transcript Simplifying Expressions to Solve Linear Equations

Equations and Inequalities
Copyright © Cengage Learning. All rights reserved.
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Section
2.3
Simplifying Expressions to Solve
Linear Equations in One Variable
Copyright © Cengage Learning. All rights reserved.
Objectives
1. Simplify an expression using the order of
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operations and combining like terms.
22. Solve a linear equation in one variable
requiring simplifying one or both sides.
33. Solve a linear equation in one variable that is
an identity or a contradiction.
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1.
Simplify an expression using the order
of operations and combining like terms
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Simplifying Expressions
The number part of each term is called its coefficient.
• The coefficient of 7x is 7.
• The coefficient of –3xy is –3.
• The coefficient of y2 is the understood factor of 1.
• The coefficient of 8 is 8.
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Simplifying Expressions
Like Terms
Like terms, or similar terms, are terms with the same
variables having the same exponents.
Like terms:
• 3x and 5x
• 9x2 and –3x2
Unlike terms:
• 4x and 5x2
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Simplify an expression using the order of operations
and combining like terms
Using the distributive property of algebra, combine the
terms in 3x + 5x and 9xy2 – 11xy2 .
3x + 5x = (3 + 5)x
expressions with
like terms = 8x
9xy2 – 11xy2 = (9 – 11)xy2
expressions with
= –2xy2
like terms
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Simplifying Expressions
Combining Like Terms
To combine like terms, add their coefficients and keep the
same variables and exponents.
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Your Turn
Simplify: 3(x + 2) + 2(x – 8)
Solution:
To simplify the expression, we will use the distributive
property to remove parentheses and then combine like
terms.
3(x + 2) + 2(x – 8)
= 3x + 3  2 + 2x + 2  (–8)
= 3x + 6 + 2x – 16
Use the distributive property
to remove parentheses.
3  2 = 6 and 2  8 = 16.
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Your Turn
= 3x + 2x + 6 – 16
= 5x – 10
cont’d
Use the commutative property of addition:
6 + 2x = 2x + 6.
Combine like terms.
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Simplifying Expressions
Comment
You a) simplify expression and then
b) solve and equation
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2.
Solve a linear equation in one variable
requiring simplifying one or both sides
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Solving an Equation
To solve a linear equation in one variable, we must isolate
the variable on one side.
Solving Equations
1. Clear the equation of any fractions or decimals.
2. Use the distributive property to remove any grouping
symbols.
3. Combine like terms on each side of the equation.
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Solving an Equation
4. Undo the operations of addition and subtraction to collect
the variables on one side and the constants on the other.
5. Combine like terms and undo the operations of
multiplication and division to isolate the variable.
6. Check the solution in the original equation.
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Your Turn
Solve: 3(x + 2) – 5x = 0.
Solution:
To solve the equation, we will remove parentheses,
combine like terms, and solve for x.
3(x + 2) – 5x = 0
3x + 3  2 – 5x = 0
3x – 5x + 6 = 0
Use the distributive property
to remove parentheses.
Use the commutative property
of addition and simplify.
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Your Turn -- Solution
–2x + 6 = 0
–2x + 6 – 6 = 0 – 6
–2x = –6
cont’d
Combine like terms.
Subtract 6 from both sides.
Combine like terms.
Divide both sides by –2.
x=3
Simplify.
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Your Turn – Solution
cont’d
Check:
3(x + 2) – 5x = 0
3(3 + 2) – 5  3 ≟ 0
35–53≟0
15 – 15 ≟ 0
0=0
Substitute 3 for x.
Perform the operation inside the
parentheses.
Multiply.
True.
Since the solution 3 checks, the solution set is {3}.
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3.
Solve a linear equation in one variable
that is an identity or a contradiction
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Solving an Equation
Conditional Equation:
Given: 3x + 1 = 0
is true only when: x = -1/3
Identity:
Given: 2x + x = 3x
Contradiction:
Given: x = x + 1
is true for all R
is never true--no solution
Solution set = Ø (Empty set)
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Solving an Equation
Three possible types of equations.
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Your Turn
Solve: 3(x + 8) + 5x = 2(12 + 4x).
Solution:
To solve this equation, we will remove parentheses,
combine terms, and solve for x.
3(x + 8) + 5x = 2(12 + 4x)
3x + 24 + 5x = 24 + 8x
8x + 24 = 24 + 8x
Use the distributive property to
remove parentheses.
Combine like terms.
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Your Turn – Solution
8x + 24 – 8x = 24 + 8x – 8x
24 = 24
cont’d
Subtract 8x from both sides.
Combine like terms.
Since the result 24 = 24 is true for every number x, every
number is a solution of the original equation.
This equation is an identity. The solution set is the set of
real numbers,
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