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Exponential Decay
Negative Exponents
Multiplying and Dividing Fractions 4
Evaluating Expressions Involving Fractions
The Cartesian Coordinate System
Adding and Subtracting Fractions with Like Denominators
Solving Absolute Value Inequalities
Multiplying Special Polynomials
FOIL Method
Inequalities
Solving Systems of Equations by Graphing
Graphing Compound Inequalities
Solving Quadratic Equations by Completing the Square
Addition Property of Equality
Square Roots
Adding and Subtracting Fractions
The Distance Formula
Graphing Logarithmic Functions
Fractions
Dividing Mixed Numbers
Evaluating Polynomials
Power of a Product Property of Exponents
Terminology of Algebraic Expressions
Adding and Subtracting Rational Expressions with Identical Denominators
Solving Exponential Equations
Factoring The Difference of 2 Squares
Changing Fractions to Decimals
Solving Linear Equations
Using Patterns to Multiply Two Binomials
Completing the Square
Roots of Complex Numbers
Methods for Solving Quadratic Equations
Conics in Standard Form
Solving Quadratic Equations by Using the Quadratic Formula
Simplifying Fractions 2
Exponential Notation
Exponential Growth
The Cartesian Plane
Graphing Linear Functions
The Slope of a Line
Finding Cube Roots of Large Numbers
Rotating Axes
Common Mistakes With Percents
Solving an Equation That Contains a Square Root
Rational Equations
Properties of Common Logs
Composition of Functions
Using Percent Equations
Solving Inequalities
Properties of Exponents
Graphing Quadratic Functions
Factoring a Polynomial by Finding the GCF
The Rectangular Coordinate System
Adding and Subtracting Fractions
Multiplying and Dividing Rational Expressions
Improper Fractions and Mixed Numbers
Properties of Exponents
Complex Solutions of Quadratic Equations
Solving Nonlinear Equations by Factoring
Solving Quadratic Equations by Factoring
Least Common Multiples
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Solving Exponential Equations
Solving Linear Equations
Multiplication Property of Equality
Multiplying Mixed Numbers
Multiplying Fractions
Reducing a Rational Expression to Lowest Terms
Literal Numbers
Factoring Trinomials
Logarithmic Functions
Adding Fractions with Unlike Denominators
Simplifying Square Roots
Adding Fractions
Equations Quadratic in Form
Dividing Rational Expressions
Slopes of Parallel Lines
Simplifying Cube Roots That Contain Variables
Functions and Graphs
Complex Numbers
Multiplying and Dividing Fractions 1
Composition of Functions
Intercepts of a Line
Powers
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Multiplying Two Numbers with the same Tens Digit and whose Ones Digits add up to 10
Factoring Trinomials
Exponents and Polynomials
Decimals and their Equivalent Fractions
Negative Integer Exponents
Adding and Subtracting Mixed Numbers
Solving Quadratic Equations
Theorem of Pythagoras
Equations 1
Subtracting Fractions
Solving Quadratic Equations by Graphing
Evaluating Polynomials
Slope
Angles and Degree Measure
   
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Solving Equations

Solving equations is a recurring theme in much of mathematics. We will study some of the principles used to solve equations and then use equations to solve applied problems.

Solving equations is essential for problem solving in algebra. In this section, we study two of the most important principles used for this task.

Equations and Solutions

We have already seen that an equation is a number sentence stating that the expressions on either side of the equals sign represent the same number. Some equations, like 3 + 2 = 5 or 2x + 6 = 2(x + 3) are always true and some, like 3 + 2 = 6 or x + 2 = x + 3 are never true. In this text, we will concentrate on equations like 3x + 6 = 17 or 7x = 141 that are sometimes true, depending on the replacement value for the variable.

Solution of an Equation: Any replacement for the variable that makes an equation true is called a solution of the equation. To solve an equation means to find all of its solutions.

To determine whether a number is a solution, we substitute that number for the variable throughout the equation. If the values on both sides of the equals sign are the same, then the number that was substituted is a solution.

Example 1

Determine whether 7 is a solution of x + 6 = 13

Solution

We have

Writing the equation.

Substituting 7 for x.

Note that 7, not 13, is the solution.

Since the left-hand and the right-hand sides are the same, 7 is a solution.

Example 2

Determine whether 19 is a solution of 7x = 141.

Solution

We have

Writing the equation.

Substituting 19 for x.

The statement 133 = 141 is false.

Since the left-hand and the right-hand sides differ, 19 is not a solution.

The Addition Principle

Consider the equation x = 7.

We can easily see that the solution of this equation is 7. Replacing x with 7, we get 7 = 7, which is true.

Now consider the equation x + 6 = 13.

In Example 1, we found that the solution of x + 6 = 13 is also 7. Although the solution of x = 7 may seem more obvious, the equations x + 6 = 13 and x = 7 are equivalent.

Equivalent Equations: Equations with the same solutions are called equivalent equations.

There are principles that enable us to begin with one equation and end up with an equivalent equation, like x = 7, for which the solution is obvious. One such principle concerns addition. The equation a = b says that a and b stand for the same number. Suppose this is true, and some number c is added to a. We get the same result if we add c to b, because a and b are the same number.

The Addition Principle: For any real numbers a, b, and c, a = b is equivalent to a + c = b + c.

To visualize the addition principle, consider a balance similar to one a jeweler might use. (See the figure.) When the two sides of the balance hold quantities of equal weight, the balance is level. If weight is added or removed, equally, on both sides, the balance will remain level.

When using the addition principle, we often say that we “add the same number to both sides of an equation”. We can also “subtract the same number from both sides”, since subtraction can be regarded as the addition of an opposite.

Example 3

Solve: x + 5 = -7.

Solution

We can add any number we like to both sides. Since is the opposite, or additive inverse, of 5, we add to each side:

x + 5 = -7

x + 5 - 5 = -7 - 5 Using the addition principle: adding -5 to both sides or subtracting 5 from both sides.

x + 0 = -12 Simplifying: x + 5 - 5 = x + 5 + (-5) = x + 0.

x = -12 Using the identity property of 0.

It is obvious that the solution of x = -12 is the number -12. To check the answer in the original equation, we substitute.

Check:

The solution of the original equation is -12.

In Example 3, note that because we added the opposite, or additive inverse, of 5, the left side of the equation simplified to x plus the additive identity, 0, or simply x. These steps effectively replaced the 5 on the left with a 0. When solving x + a = b for x, we simply add -a to (or subtract a from) both sides.

Example 4

Solve: -6.5 = y - 8.4

The variable is on the right side this time. We can isolate y by adding 8.4 to each side:

-6.5 = y - 8.4 This can be regarded as -6.5 = y + (- 8.4)

-6.5 + 8.4 = y - 8.4 + 8.4 Using the addition principle: Adding 8.4 to both sides “eliminates” -8.4 on the right side.

1.9 = y Sice y - 8.4 + 8.4 = y + (-8.4) + 8.4 = y + 0 = y

Check:

The solution is 1.9.

Note that the equations a = b and b = a have the same meaning. Thus, -6.5 = y - 8.4 could have been rewritten as y-8.4 = -6.5.

Example 5

Solve:

Solution

We have

Adding to both sides.

Multiplying by 1 to obtain a common denominator

The check is left to the student. The solution is.

The Multiplication Principle

A second principle for solving equations concerns multiplying. Suppose a and b are equal. If a and b are multiplied by some number c , then ac and bc will also be equal.

The Multiplication Principle: For any real numbers a , b , and c , with c 0, a = b is equivalent to a.c = b.c.

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