Algebra/Chapter 3/Equations

Algebra

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Algebra Chapter 3: Solving Equations Section 1: What are Equations?

Inverses of Numbers

3.1: What are Equations?

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Balancing variables

Mathematics began with the concept of counting, but it grew because of the idea of equivalence. Tally marks or counters are equivalent to a value depending on the concept being expressed. For instance a shepherd could use a bag of rocks to make sure that he brought home the same number of sheep he left with. The idea of equivalence as shown with an equal sign allows us to state "is the same as".

An expression is a statement that is well formed. The neolithic shepherd used his bag of rocks to create expressions for managing his herd. For instance, the shepherd hoped that the size of his flock remained constant between the time he took his flock out to graze and the time that he returned them home. When he used the bag of rocks the shepherd used two expressions: # Sheep and # Rocks to create the equation # Sheep = # Rocks. Mathematics describes the operations the shepherd performed on the expressions in order to keep the equation in balance. For instance, when the shepherd used his bag of rocks to count his flock at night he might put his rocks in a bowl next to the gate on his pen. Each time a sheep went into the pen the shepherd would move a rock from the bag into the bowl. If rocks were left over in the bag, then the shepherd knew he needed to return to the field to find his lost sheep.

Expressions

An expression is the arrangement of mathematical symbols denoting values and operations, while the equation is a form of an expression that indicates that two values are equal. While equations are generally expressions, you will discover that some expressions are not equations. Expressions in the form of functions or inequalities will be more apparent later in the book.

When you simplify an expression, you are performing simple arithmetic steps until you receive the simplest answer, which is usually an integer or fraction.

Equations

A balance (scale).

An equation is just like a balance. A balance is a machine that compares whether two quantities are the same or not the same. In mathematics, two quantities balanced translates to "having the same value." An equation is true if the two sides are balanced and false if the two sides are not balanced. As 5=5. This is a very simple equation. We can use variables too(almost used in mathematics)e.g. x=5. This is same as while using a balancing machine, we use some quantity on one side and some matter on the other which is to be measured. As an example we know that 5 is equal to 5 and x is equal to x(whatever its value is). The simplest example of a false equation is that 5 is not equal to 20.

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Remember from the previous section, a variable is a symbol, usually a letter, that stands for a number. We'll use the letters $x$ and $y$ most often. If $x=5$ is true, then any time I use $x$, you can replace $x$ by 5. We're saying $x$ and 5 have the same value, but different appearance. In the equation $x=5$, the equal sign is like the balance. If $x=5$ is true, we know that whenever we see an $x$ we can substitute the number 5 if that helps us understand. Even more important, we can check if a given number substituted for a variable in an equation makes that equation true.

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Frequently we don't know what the value of a variable is that would make the equation true, and we need to find out! Finding out the values that make an equation true is called Solving an Equation, which we will do without guessing in the next section.

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Practice Problems

Conceptual Questions

Problem 3.1 (Explaining Equations) In no more than one paragraph, explain in your own words what an "equation" is. In your definition, discuss an equation's purpose, as well as what a solution of an equation means.

Problem 3.2 (Anatomy of an Equation) Given the statement x + 6 = 12, answer the following.

a. Is this statement true or false?
b. Does x=5 make the statement true?
c. Does x=6 make the statement true?

Problem 3.3 (Identifying Equations) Determine which of the following are equations.

a. x + 4
b. a - 64 = 45
c. 7 < 8
d. a + 6 > 9
e. 1 + 7 = 8

Problem 3.4 (Special Solution Sets) Make up an equation whose solution is the null set, and explain why this is the solution set. After this, make up an equation whose solution is the set of all real numbers, and explain why this is the solution set.

Problem 3.5 (Equivalent Equations) Are the statements 9x + 5 = 4 and 4 = 9x + 5 the same equation? Explain your reasoning.

Exercises

Problem 3.6 (Checking Solutions) Check if the given value(s) is/are solutions to the following equations.

Problem 3.7 (Equation Diagrams) Write the equation that best represents the following diagrams.

Problem 3.8 (Scales) Look at the diagrams below. How many circles would you need to balance the third scale?

Problem 3.9 (Scales) Which is heavier? A red triangle or a blue square?