Thinking Mathematically

1
Thinking Mathematically

• Algebra Equations and Inequalities
• 6.2 Solving Linear Equations

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Solving Linear Equations
Two algebraic expressions connected with an
equal sign is called an equation. When there
are no exponents for the variables (other than
one) the equation is called linear.
Solving an equation means finding all of the
numbers for the variables that make the equal
sign true. For equations with only one variable,
the solution is the set of all of those numbers.
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The Addition Property of Equality

• The same real number (or algebraic expression)
  may be added to both sides of an equation without
  changing the solution set. This can be expressed
  symbolically as follows
• If a b, then a c b c.

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The Subtraction Property of Equality

• The same real number (or algebraic expression)
  may be subtracted from both sides of an equation
  without changing the solution set.
• If a b, then a c b c.

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SolvingLinear Equations
The rules of equality can be used to solve a
linear equation. Any number or variable can be
added or subtracted from both sides. The goal
is to isolate the variable on one side of the
equal sign.
Exercise Set 6.2 3 x 5 -12
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The Multiplication Property of Equality

• The same nonzero real number (or algebraic
  expression) may multiply both sides of an
  equation without changing the solution set.
• If a b and c ? 0, then ac bc.

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The Division Property of Equality

• Both sides of an equation may be divided by the
  same nonzero real number (or algebraic
  expression) without changing the solutions set.
• If a b and c ? 0, then a/c b/c.

8
SolvingLinear Equations
The rules of equality can be used to solve a
linear equation. Both sides of an equation may
be multiplied or divided by any (non-zero) number
or variable. The goal is to isolate the
variable on one side of the equal sign.
Exercise Set 6.2 7 5x 45
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Solving a Linear Equation

1. Simplify the algebraic expression on each side.
2. Collect all the variable terms on one side and
   all the constant terms on the other side.
3. Isolate the variable and solve.
4. Check the proposed solution in the original
   equation.

Exercise Set 6.2 19, 23, 41 14 5x -41 5x
(2x 10) 35 6 -4(1 x) 3(x 1)
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Misc.

• Solving linear equation with fractions
• Exercise Set 6.2 47

• Special cases no solution, all real numbers
• Exercise Set 6.2 75, 81
• 2(x 4) 4x 5 2x 3
• 4(x 2) 1 7x 3(x 2)

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Thinking Mathematically

• Algebra Equations and Inequalities
• 6.2 Solving Linear Equations