Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

R.5 The Basics of Equation Solving

Solve linear equations. Solve quadratic equations. Solve a formula for a given letter.

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Equations

An equation is a statement that two expressions are equal.

To solve an equation in one variable is to find all the values of the variable that make the equation true.

Each of these numbers is a solution of the equation. The set of all solutions of an equation is its solution set. Equations that have the same solution sets are called equivalent equations.

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Equations

A linear equation in one variable is an equation that is equivalent to one of the form ax + b = 0, where a and b are real numbers and a 0.

A quadratic equation is an equation that is equivalent to one of the form ax2 + bx + c = 0, where a, b and c are real numbers and a 0.

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Equation- Solving Principles

For any real numbers a, b, and c,The Addition Principle: If a = b is true, then

a + c = b + c is true. The Multiplication Principle: If a = b is true, then

ac = bc is true.The Principle of Zero Products: If ab = 0 is true, then

a = 0 or b = 0, and if a = 0 or b = 0, then ab = 0.

The Principle of Square Roots: If x2 = k, then or .x k x k

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Examples

Solve. 8x = 2(18 – 2x)

8x = 36 – 4x12x = 36 x = 3

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ExamplesSolve.

x2 – 8x = – 16 x2 – 8x + 16 = 0

(x – 4 )(x – 4) = 0 x – 4 = 0 or x – 4 = 0 x = 4 or x = 4

There is only one solution, 4.

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Examples

Solve.

2 4 12 0 x

3 or 3x x

2 4 12x 2 3x

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Formulas

A formula is an equation that can be used to model a situation.

We have used the motion formula, d = r • t.

We have used the simple-interest formula, I = Prt.

We can use the equation-solving principles to solve a formula for a given variable.

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ExampleSolve P = 2l + 2w for l.

Solution:P 2l 2w

P 2w 2l

P 2w2

l

l P 2w

2The formula can be used to determine a

rectangle’s length if we know its perimeter and width.