Download Sec 6.2 Notes

Math in Our World
Section 6.2
Solving Linear Equations
Learning Objectives
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Decide if a number is a solution of an equation.
Identify linear equations.
Solve general linear equations.
Solve linear equations containing fractions.
Solve formulas for one specific variable.
Determine if an equation is an identity or a
contradiction.
Equations
An equation is a statement that two algebraic
expressions are equal.
A solution of an equation is a value of the variable that
makes the equation a true statement when substituted into
the equation. Solving an equation means finding every
solution of the equation. We call the set of all solutions the
solution set, or simply the solution of an equation.
For example, x = 2 is one solution of the equation x2 – 4 = 0,
because (2)2 – 4 = 0 is a true statement. But x = 2 is not the
solution, because x = – 2 is a solution as well. The solution set
is actually {– 2, 2}.
Expressions vs. Equations
Note the difference between the two; equations
contain an equal sign and expressions do not.
EXAMPLE 1
Identifying Solutions of an
Equation
Determine if the given value is a solution of the
equation.
(a) 4(x – 1) = 8; x = 2
(b) x + 7 = 2x – 1; x = 8
(c) 2y2 = 200; y = – 10
Linear Equations
A linear equation does not break the
following rules:
*variables do not contain exponents greater
than 1
*variables are not square rooted
*variables do not appear in the denominator
of a fraction
*variables are not multiplied together
EXAMPLE 2
Identifying Linear Equations
Determine which of the equations below are linear
equations.
Solving Linear Equations
One-step Equations
*Addition/Subtraction Properties
*Multiplication/Division Properties
EXAMPLE 3
Solving Linear Equations
Solve each equation using the Addition and
Subtraction Property, and check your answer.
(a) x – 5 = 9
(b) y + 30 = 110
EXAMPLE 4
Solving Linear Equations
Solve each equation using the Multiplication and
Division Property, and check your answer.
Solving Linear Equations
Multi-step Equations
Procedures for Solving Linear Equations
Step 1 Simplify the expressions on both sides of the
equation by distributing and combining like terms.
Step 2 Get the term with the variable by itself using
the addition or subtraction properties
Step 3 Use the multiplication or division properties to
solve for the variable.
EXAMPLE 5
Solving a Linear Equation
Solve the equation 5x + 9 = 29.
EXAMPLE 6
Solving a Linear Equation
Solve the equation 6x – 10 = 4x + 8.
EXAMPLE 7
Solving a Linear Equation
Solve the equation 3(2x + 5) – 10 = 3x – 10.
Solving Equations
Containing Fractions
There’s a simple procedure that will turn any
equation with fractions into one with no fractions at
all. You just need to find the Least Common
Denominator of all fractions that appear in the
equation, and multiply every single term on each
side of the equation by the LCD.
If there are any fractions left after doing so, you
made a mistake!
EXAMPLE 8
Solving Linear Equations
Containing Fractions
Solve the equation:
EXAMPLE 9
Solving Linear Equations
Containing Fractions
2 x  3 5x x
Solve the equation:
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 4
3
2
2
Solve formulas for one specific variable.
EXAMPLE 10
Solving a Formula in
Electronics for One Variable
Solve the formula
EXAMPLE 11
Finding a Formula for
Temperature in Celsius
The formula F = 95 C + 32 gives the Fahrenheit
equivalent for a temperature in Celsius.
Transform this into a formula for calculating the
Celsius temperature C.
Contradictions and Identities
A contradiction is an equation with no solution.
An identity is an equation that is true for any
value of the variable for which both sides are
defined.
When you solve an equation that is an identity, the
final equation will be a statement that is always true.
In a contradiction the final equation will be a statement
that is false.
EXAMPLE 12
Recognizing Identities and
Contradictions
Indicate whether the equation is an identity or a
contradiction, and give the solution set.
(a) 3(x – 6) + 2x = 5x – 18
(b) 6x – 4 + 2x = 8x – 10
Classwork
p. 286-287: 7-71 eoo, 75, 77, 83