Equations
Vocabulary
The following terms are important to understand when working with algebraic equations.
| Term | Definition |
|---|---|
| Mathematical Term | A mathematical expression that does not contain +, - or equality symbols. |
| Coefficient | The number being multiplied to a variable in a mathematical term. |
| Constant Term | A mathematical term with no variable or exponents. |
| Like Terms | Two or more mathematical terms having the same variable and the same exponent. |
| Equation | Two mathematical expressions that have been set equal to each other, such as $2 + 5 = 7$. |
| Equivalent Equations | Two equations that look different, but are mathematically identical. |
| Inequality | An equation in which the two sides are not equal. |
| Variable | A symbol (or placeholder) for a value we don’t yet know. |
| Exponent | The number of times the base of a power should be multiplied. |
| Literal Equation | Equations made up of alphabet variables, such as $E = MC^2$. |
| Inverse Operation | An operation that reverses another operation, such as addition and subtraction. |
| Ratio | TWo values that have been set in relation to each other. |
| Proportion | Two ratios that are set in relation to each other. |
| Rate | A proportion in which one value has been set to 1. |
| Cross Multiplication (or Cross Product) | Multiplying the numerators of fraction with each other’s denominators. |
| General Mathematical Principle | A core principle of mathematics. |
| Addition Property of Equality | If we add or subtract the same number to both sides of an equation, the sides remain equal. |
| Multiplication Property of Equality | If we multiply or divide the same number to both sides of an equation, the sides remain equal. |
| Commutative Property of Addition | The order in which numbers are added does not change the sum. (This property does not apply to subtraction.) |
| Commutative Property of Multiplication | The order in which numbers are multiplied does not change the sum. (This property does not apply to division.) |
| Associative Property of Addition | The order in which numbers are grouped when adding does not change the sum. (This property does not apply to subtraction.) |
| Associative Property of Multiplication | The order in which numbers are grouped when multiplying does not change the product. (This property does not apply to division.) |
| Distributive Property | |
| Identity Property of Addition | When a number is added to zero, the result is the original number. |
| Identity Property of Multiplication | When a number is multiplied by 1, the result is the original number. |
| Justification | The mathematical principle used to transform an equation into another form. |
| Solution | The answer to a mathematical problem. |
| Solution Set | The set of all possible solutions to a mathematical problem. |
About
Logical reasoning is central to all mathematics, and is at the heart of algebraic thinking. A proper justification for each step in the solution of an equation is not simply a description of what was done (“I added 4 to both sides.”), but rather a general principle, such as The Addition Property of Equality.
Decimal System
Tips
- When adding or subtracting decimal numbers, always line up the decimal points.
- When multiplying decimal numbers, count all the digits to the right of both decimal points, and move the decimal point in the answer that many places to the left. There is no need to line up the decimal points when multiplying.
- When dividing decimal numbers, always move the decimal point in the divisor to the right as many places as needed to make it a whole number. Then move the decimal point in the dividend the same number of places, adding zeroes if needed.
The Decimal Number System, also called Hindu-Arabic number system, is a positional numeral system that uses 10 as the base and requires 10 different symbols which are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. It also uses a dot (the decimal point) to represent fractions. In this system, the numerals used to show a value take different place values depending upon position. In a base-10 system the number 543.21 represents the sum:
Squares and Square Roots
A headlong plunge into the mighty story of perfect squares and their roots.Area in the Coordinate Plane
Finding the area of polygons defined by linear equations.Danish Hearts
History
The first Danish Hearts were probably made by mothers and children, but that is lost to history. What is still known is that Hans Christian Andersen, the great Danish author of folk-tales, created one in 1860.
Danish hearts (Julehjerter, literally “Yule Hearts”) soon became a Winter Solstice tradition in Denmark and other Scandinavian countries. Families often have a “cut and paste” day on which they create crafts from paper to decorate their homes. They fill these baskets with candies and nuts to hang on trees or give as presents.
Metric Measurements
Place Value
The Metric System of Measures takes advantage of the fact that in the Decimal Number System it’s easy to multiply and divided by groups of 10. In the decimal system, the place (or position) of a symbol determines its value (the actual numeric quantity). In the below examples, the only thing that changes is the position of the symbol $1$, but the value of that symbol changes by multiples of $10$ from very large to very small.
Metric to Us Measurement Conversions
| US | Metric | |
|---|---|---|
| Length | ||
| 1 inch (in) | 2.54 centimeters (cm) | |
| 1 foot (ft) | 30.48 centimeters | |
| 1 yard (yd) | 0.914 meters (m) | |
| 1 mile (mi) | 1.609 kilometers (km) | |
| Weight | ||
| 1 ounce | 28.3 grams | |
| 1 pound | 0.45 kg | |
| Area | ||
| 1 in2 | 6.45 cm2 | |
| 1.196 yd2 | 1 m2 | |
| 2.47 acres | 1 ha | |
| Volume | ||
| 1 in3 | 16.39 mL | |
| 1.06 qt | 1 liter | |
| 1 gallon | 3.79 liters | |
| 1 m3 | 35.31 ft3 | |
| 1 quart | 0.95 liters (L) | |
| Temperature | $ C^\circ \text{ to } F^\circ$ | $ F^\circ \text{ to } C^\circ$ |
| $ F^\circ = \frac{9}{5}C + 32$ | $C^\circ = \frac{5}{9}(F - 32)$ |
Prime Numbers
What is the Prime Number?
Prime numbers are all integers greater than 1 that have only two factors, the number itself and 1. For example, the only factors of 3 are 1 and 3, so 3 is prime.
The smallest prime number is 2 because it is only divisible by itself and 1. The number 2 also happens to be the only even prime number.
A number that has more than two factors is called a composite number. Such numbers can be made by combining more than two factors. For example, $2 \times 2 \times 3 = 12$, therefore 12 iS a composite number.
