While it’s not obvious, learning equations are important to our daily lives. Without it, we would not have computers, GPS, satellite TV, and other inventions that make modern society what it is.
Some of the equations that changed the way we live are The Pythagorean Theorem, The fundamental theorem of calculus, and Newton’s universal law of gravitation, just to name a few.
The equation is essential to medicine, economics, computer science, engineering, and many more. Read on if you want to learn more about this dynamic concept.
Types of Equations – Algebraic
A cubic equation is a polynomial equation whereby the highest sum of exponents of the variables in any term is equal to three. In other words, it is an equation involving a cubic polynomial; i.e., one of the forms. It has the following form:
ax3 + bx2 + cx + d = 0 where a ≠ 0
Exponential equations have variables in the place of exponents, and can be solved using this property: axax = ayay => x = y. Examples include the following:
- 4x = 0
- 8x = 32
- ab = 0 (where “a” is base and “b” is the exponent)
Irrational Polynomial Equation
Irrational polynomial equations are those equations with at least a polynomial under the radical sign.
Linear equations are those where each term is either a constant or the product of a single variable and a constant. If there are two variables, the graph of the linear equation is always going to be a straight line. As a general rule, a linear equation looks like this:
y = mx + c, m ≠≠ 0
In this example, m is known as slope and c represents that point on which it cut the y axis.
In linear equations with different variables:
The equation with only one variable: an equation that has only one variable. Examples include the following:
- 8a – 8 = 0
- 9a = 72
The equation has two variables: an equation that has only two types of variables. Examples include the following:
- 9a + 6b – 82 = 0
- 7x + 7y = 12
- 8a – 8d = 74
The equation that has three variables: this is an equation with only three types of variables in the equation. Examples include the following:
- 13a – 8b + 31c = 74
- 5x + 7y – 6z = 12
- 6p + 14q – 74 + 82 = 0
These are equations whereby the unknown is always affected by a logarithm.
Polynomial equations contain either variables or indeterminates and coefficients. These are involved in operations such as addition, subtraction, multiplication, and non-negative integer exponents. Examples include the following:
- ax + by + c = 0ax + by + c = 0 with degree = 1 and two variables
- ax2 + bx + c = 0ax2 + bx = c = 0 with degree = 2 and one variable
- ax + b = 0 with degree = 1 and one variable
- axy + c = 0axy + c = 0 with degree = 2 and two variables
A quadratic equation is a second-degree equation whereby one variable contains the variable that has an exponent of two. An example and the general form is shown below.
ax2 + bx + c = 0, a ≠≠ 0
Other examples include:
- 5a2 – 5a = 35
- 8x2 + 7x – 75 = 0
- 4y2 + 14y – 8 = 0
Quartic equations are equations of the fourth degree and an equation that equates a quartic polynomial to zero, using this form:
f(x) = ax4 + bx3 + cx2 + dx + e = 0 where a ≠ 0
The derivative of a quartic function is a cubic function.
A quintic equation is a polynomial equation in which five is the highest power of the variable. The formula used is:
ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0
Examples include the following:
- x^5 + x^3 + x
- y^5 + y^4 + y^3 + y^2 + y + 1
Radical equations are those that have a maximum exponent on the variable that is 12 and which have more than one term. It can also be said that a radical equation is one whereby the variable is lying inside a radical symbol, usually in the form of a square root. Examples include the following:
- + 10 = 26
- + x – 1
A rational equation involves rational expressions.
Transcendental equations are equations that include transcendental functions. Exponential equations are examples of transcendental equations.
Put simply, trigonometric equations are those equations that involve trigonometric functions, usually of unknown angles such as cos B = ½.
An example of a trigonometric equation is found here:
More samples of trigonometric equations can be found here.
Other examples of algebraic equations can be found here.
Types of Equations – Geometric
Figures – Formulas for Volume (V) and Surface Area (SA)
- V = Bh = area of base x height
- SA = sum of the areas of the faces
- V = lwh = length x width x height
- SA = 2lw + 2hw + 2lh
- = 2(length x width) + 2(height x width) + 2(length x height)
Right Circular Cone
- V = Bh = x area of base x height
- SA = B + C
- = area of base + ( x circumference of base x slant height)
Right Circular Cylinder
- V = Bh = area of base x height
- SA = 2B + Ch = (2 x area of base) + (circumference x height)
- V = 3 = x x cube of radius
- SA = 4 2 = 4 x x square of radius
- V = Bh = x area of base x height
- SA = B + P
- = area of base + ( x perimeter of base x slant height)
Shapes – Formulas for Area (A) and Circumference (C)
A = 2 = x square of radius
C = 2 r = 2 x x radius
C = = x diameter
A = bh = base x height
A = lw = length x width
A = (b1 + b2)h = x sum of bases x height
A = bh = x base x height
Equations of a Line
Ax + By = 0 where A and B are not zero
y – y1 = m(x – x1) where m = slop, (x1, y1) = point on line
y = mx + b or y = b + mx where m = slope b = y – intercept
Area of circle: 2 ( = 3.14 approximately)
Area of a rectangle: length x height
Area of square: length2 (l x l)
Area of a triangle: ½ x length x height
Circumference of a circle: 2 ( x diameter)
Volume of cone: 1/3 x area of base x height; 1/3 x (d/2)2 x h
Volume of cylinder: area of base x height; (d/2)2 x h
Volume of rectangle prism: length x height x depth
Types of Chemical Reactions
Combination Chemical Reactions
In this reaction, one product is formed by two or more reactants.
Moreover, more than one product can be formed in a combination chemical reaction, depending on the conditions or the relative amounts of the reactants.
Combustion Chemical Reactions
Combustion chemical reactions happen when a compound, which usually contains carbon, combines with the oxygen gas found in the air. This process is called burning, as heat is the most important product when it comes to most combustion chemical reactions.
Propane is part of the compounds called hydrocarbons, which are compounds that are composed of carbon and hydrogen only. Heat is the result of this reaction. In addition, combustion chemical reactions are a type of redox chemical reaction as well.
Decomposition Chemical Reactions
In reality, decomposition chemical reactions are the exact opposite of combination reactions. With decomposition reactions, a single compound breaks down into two or more substances that are simpler in nature, usually either compounds and/or elements.
Double-Displacement Chemical Reactions
Single-displacement reactions involve only one chemical species being displaced; however, in double-displacement reactions – also called metathesis reactions – two species, usually ions, are displaced.
More often than not, these types of chemical reactions occur in a solution, whereby either water (neutralization reactions) or an insoluble solid (precipitation reactions) are formed.
Neutralization Chemical Reactions
This is another type of double-displacement chemical reaction that occurs between a base and an acid. Called a neutralization reaction, this double-displacement type of chemical reaction forms water. Examples include the following:
The mixing of sodium hydroxide (lye) and sulfuric acid (auto battery acid) is a reaction that is demonstrated as:
Polymerization Chemical Reactions
Polymerization is a process whereby monomer molecules react together in a chemical reaction, which results in the forming of polymer chains, also called three-dimensional networks. There are numerous forms of polymerization, in addition to different systems that categorize each of them. Examples include the following:
nH2C=CH2 → [-CH2CH2-]n
This equation represents the union of thousands of ethylene molecules, which results in polyethylene.
In both cellulose and starch, glucose molecules join together with the concomitant elimination of a molecule of water for every linkage that is formed. An example of this is demonstrated as:
nC6H12O6 → -[-C6H10O5-]-n + nH2O
Precipitation Chemical Reactions
Mixing a solution of silver nitrate with a solution of potassium chloride results in a white insoluble. Any time you form an insoluble solid in a solution, it is called precipitation, and the white insoluble solid that is formed is called silver chloride.
Redox Chemical Reactions
Also called reduction-oxidation chemical reactions, these reactions involve electrons being exchanged.
These are also examples of other types of reactions – including combination, single-replacement, and combustion reactions – but are all redox reactions. All of them involve electrons being transferred from one chemical species to another.
Redox chemical reactions are also involved in rusting, photosynthesis, combustion, batteries, respiration, and much more.
Single-Displacement Chemical Reactions
Single-displacement reactions occur when a more active element displaces or kicks out another element that is less active from a compound. An example would be if you place some zinc metal into a copper sulfate solution, the zinc actually displaces the copper.
In this equation, the notation (aq) means that the compound is dissolved in water, which is an aqueous solution. Since zinc replaces copper in this instance, it is considered to be more active. If you were to place a piece of copper in a zinc-sulfate solution, nothing happens.
More information on chemical reactions can be viewed here.
Glossary of Algebraic Terms
Absolute Value: Refers to the distance a number is from 0.
Algebra: A type of math using mathematical symbols and the rules involved in manipulating those symbols.
Associated Law of Addition: This states that for any three numbers a, b, and c, the following is always true: (a+b)+c=a+(b+c)
Base: A number raised to a power.
Ceiling Function: Ceiling(x) is the closest integer that is greater than or equal to x.
Coefficient: This is a constant that is multiplied by an expression or a variable.
Composition: The composition of two functions, f and g, is the function f∘g that transforms x into f(g(x)).
Coordinates: A point on a two-dimensional plane is always described by a pair: (x, y). In this example, the coordinate x is given by the below-the-grid labels, and the coordinate y is given by the labels found to the left of the grid.
Cube Root: The cube root of a, which is written as 3 a, is the number whose cube is a; in other words, (3 a)3=a.
Data: A collection of measurements that are related.
Domain: The set of inputs (x-coordinates) of a function or relation.
Equation: A mathematical sentence that has an equal sign; for example, 3x+5=11.
Exponent: In power, this represents the number of times the base is multiplied by itself.
Expression: A combination of numbers and variables using arithmetic; for example, 6-x.
Factor: An expression multiplied by another expression or one that is able to be multiplied by another expression in order to produce a specific result.
Function: A relation whereby no x-coordinate is seen in more than one ordered pair (x, y). In other words, think of a function as a transformation taking each x-coordinate to its single corresponding y-coordinate.
Inequality: This is a mathematical sequence using one of the following symbols: <, >, ≤, or ≥.
Integer: A whole number, or the negative of a whole number; for example, 37 and 0 and -5 are integers, but 2.7 isn’t.
Isolate: To make a variable appear alone on one side of inequality or equation, and not happen on the other side of the inequality or equation.
Joint Frequency: This refers to the number of events that satisfy both parts of two specified criteria.
Joint Relative Frequency: This is a joint frequency that is divided by the total number of events.
Monic: A polynomial whose first or leading coefficient is 1.
Monomial: A product of numbers and variables; for example, 3x or 5x2. It is also occasionally called a term.
nth Root: An nth root of a is a number b, which has an nth power of a. In other words, bn=a.
Origin: This is the point on a coordinate plane where the x-axis and the y-axis intersect one another. It is always represented by the coordinates (0,0).
Piecewise-Defined Function: This is a function defined by different formulas at different inputs.
Point: A point is a location on the coordinate plane. It has coordinates (x, y), where x is given by the labels below the coordinate grid, and y is given by the labels to the left of a coordinate grid.
Range: A set of outputs, i.e., y-coordinates, of a function or relation.
Relation: This term refers to a set of ordered pairs, i.e., (x, y).
Relative Frequency: A frequency divided by the total number of events. It is commonly expressed as a percentage.
Sequence: A list of numbers that can be generated by some rule.
Set: An unordered collection of numbers or other objects mathematical in nature, with no repetitions.
Similar: Two geometric figures are considered similar if they have the same shape but potentially different sizes, and they have corresponding lengths that differ by a single common scale factor.
Simplify: This refers to rewriting an expression whereby it means the same thing, but it is either shorter or simpler. For example, you can simply 3x-x+6 into 2x+6.
Slope: This is a number measuring how steep a line is. It shows the amount of change in a line’s height as you go one unit to the right. For example, the slope of the line y=mx+b is the letter m.
Slope-Intercept Form: For a linear equation, the form y=mx+b, whereby b and m are constants. The numbers b and m give the slope and y-intercept of the line that is the graph in that particular section.
Solution: In either inequality or equation, the numbers can be substituted for the variable in order for that equation or inequality to be made true. If there is more than one variable in the inequality of the equation, a solution refers to a list of numbers that, when substituted for the list of variables, makes the inequality or equation true. For systems with more than one inequality or equation, the solution has to make all of the inequalities or equations true. In addition, a solution refers to a liquid mixture in chemistry.
Solution Set: This refers to all solutions to inequality, equation, or a system.
Solve: To solve is to find the solutions to inequality, equation, or a system.
Square Root: The square root of a is a number b whose square is a. In other words, b2=a. If b is a square root of a, then -b is as well.
Standard Deviation: This term refers to the square root of the variance.
Standard Form: In a linear equation, the form Ax+By=C, whereby A, B, and C are constants. For quadratic equations, either the form ax2+bx+c=0 or the form y=ax2+bx+c, where a, b, and c are constants.
Statistic: A statistic is a number that describes or summarizes data.
Statistics: Statistics is the study of data; it also refers to the methods used to summarize or describe data.
Step Function: This refers to a piecewise-defined function whereby the formula of each piece is constant; that is, it doesn’t change with x. In fact, the graph of a function looks like stair steps.
Substitution: This is the elimination of a variable in an equation or expression; it is done by replacing it with another expression that it is equal to.
System: For inequalities or equations, two or more of them are all that is required to be true.
Table: A mathematical term that involves a rectangular arrangement of columns and rows.
Term: A term is an element in a difference, sum, or sequence.
Translation: Translation is the rigid motion by a constant distance going in a single direction; that is, with no reflection or rotation.
Unit: This refers to a standard measurement; for example, an hour or a meter.
Value: This refers to a number that either an expression or a variable can equal.
Variable: A letter (for example, x) that is used to mean different numbers at different times.
Variance: The mean-squared distance of data values from their mean m. it can be computed by adding (x-m)2 for each data value x, then by dividing by the number of data values n. If you’re measuring samples from a population (for example, people’s heights), the sample’s variance is usually different than an entire population’s variance.
Vertex: This is the point where a parabola crosses its axis of symmetry, or an end of a polygon’s side, or even the corner point of an angle.
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