Equation
An equation is a mathematical statement that asserts the equality of two expressions, typically represented with an equal sign (" = ") between them. For example, an equation like "x + 3 = 7" indicates that the expression on the left is equal to the expression on the right. The history of equations dates back to ancient mathematicians such as Diophantus and later, Muhammad ibn Musa al-Khwarizmi, who contributed significantly to the development of algebra and quadratic equations. Equations comprise various components, including variables, coefficients, constants, and operators, which work together to form expressions that can be solved.
When solving equations, one must maintain balance; any operation performed on one side must be mirrored on the other side to preserve equality. Different types of equations, such as linear equations and rational equations, serve different purposes in mathematics, including graphing relationships between variables. The application of equations extends beyond theoretical mathematics; they are practical tools used in real-world scenarios, such as calculating quantities or determining patterns, like predicting growth based on time intervals. Understanding equations and their components is essential for problem-solving in mathematics and various fields of study.
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Equation
Equation means "equal" and it always refers to two things that have an equal sign " = " between them. An equation often looks something like " x + 3 = 7 " or " y – 5 = 2 " with " = " standing between the two sides and everything on the left being equal to everything on the right.
Equations were first developed by Diophantus, who wrote a book called Arithmetica (c. 150–c. 250 CE). The book dealt with number theory and showed that the purely rhetorical and symbolic form of equations could be transformed into something more like our modern, algebraic equations. Many centuries later, Muhammad ibn Musa al-Khwarizmi (c. 780–c. 850) wrote The Condensed Book on Restoration and Balancing with the first method for solving quadratic equations. His book remains the new reference on the theory of equations.
Expressions, Terms, Variables, and Operators
Equations can actually be broken down into six parts known as expressions, variables, terms, operators, coefficients, and constants. An expression involves a combination of parts. A term is already complex because it refers to either a variable, or a single number, or a combination of numbers and variables multiplied together. Examples of several different terms are x or 2x or even the long and complicated 2x (3 + y).
Terms can be broken down into smaller parts, such as variables, coefficients, constants, and operators. An operator is a symbol, such as +, ×, and ÷. Operators show an operation like addition or multiplication and is probably the easiest part of an equation to identify. A variable is also fairly easy to identify as a symbol for a number like x or y that isn’t known yet. Any number that is sitting all by itself is called a constant, and a coefficient is a number used to multiply a variable, as in 2x, which means 2 times x ; the number 2 is the coefficient.
Since a term is either a single number, or a variable, or both numbers and variables multiplied together, there can be several terms, as in 2x or 5 in an expression. An expression is a group of terms where the terms are separated by + or – sign.
Solving Equations
When solving an equation, for example, 7 – x = 3y, it is helpful to think of the equation as a balance and the equals sign (=) as the center of the balance. If something is changed on one side of the equation, the same thing has to happen on the other side. Making the same change on both sides of an equation, such as adding 5 to each side, keeps the equation balanced and, in this case, gives the new equation 12 – x = 3y + 5.
Solving an equation is the same as looking for the value of something and moving everything else to the other side of the equation. In the case of the current equation, the constant 12 needs to be moved (that is, subtracted from both sides, so that it disappears from one side and reappears on the other side as a negative number) so that –x = 3y + 5 – 12. Now simplify the equation (–x = 3y + 7). Solving for x puts that variable on one side all by itself—changing x from a negative number to a positive number and making its equivalent negative so that x = – (3y –7).
Linear Equations with Variables on Both Sides
A linear equation is an example of an equation with variables on both sides and is often used to find values on a graph. If a line is drawn on a graph with x-axis and y-axis, the slope-intercept form of an equation is used to determine the slope of the line on the graph or the y-intercept. This form of an equation looks like y = mx + b where m is the slope and b is y-intercept, or the place where the line intercepts the y-axis. This point b is also the point where x = 0 so the information can be used to solve for different variables in the linear equation.
An example of a linear equation is y = 5x – 1. To find the y-intercept, after already being given the slope (m) and a point (x, y), it is fairly easy to find the y-intercept (b). Using the form y = mx + b, the y-intercept can be identified in the above equation as –1.
Simple Rational Equations
A simple rational equation is an expression that is also understood as the ratio of two polynomials. A ratio is when one thing is divided by another, and a polynomial is an expression that has both variables and coefficients, for example, 2x – 7. The polynomial is simple because it never involves more than the operations of addition, subtraction, multiplication, and non-negative integer exponents such as x2. A negative integer exponent variable like x-2 could never be used in a polynomial. Instead, an example of a polynomial that could be used in a simple rational expression is x3 – 3x + 7. This becomes an expression used in rational equations when the polynomial is divided by another polynomial (in this case, 8x). All of this can be made equal to something else so that the final simple rational equation can be seen in Figure 1.
Equations from Tables
Understanding an equation from a table requires that the table be viewed as a series of points on a graph. The points create a line on the graph and the equation represents the different points along that line. Figure 2 represents a plant’s growth over 8 hours and the time is expressed in hours and the height of the plant in inches as it grows. Since this activity can be expressed according to an x-axis and y-axis then this table can also be expressed as a linear equation in slope intercept form y = mx + b.
A graph isn’t needed to change these numbers into a linear equation representing a line on a graph. Each point on the line has an x-coordinate and a y-coordinate representing time and height. The first point is listed as 0 hours in time (x) and 6 inches in height (y). The next point is 2 hours time (x) and 10 inches height (y). Taking any two points on the line, such as (0, 6) and (2, 10), the slope can be found by finding the change in x (2) divided by the change in y (4).
After finding the slope, the constant 2 can be substituted into the variable m for the slope, and slope intercept form of the equation y = mx + b becomes y = 2x + b. Since the line intersects the y-axis at (0, 6) where b = 6, the equation then becomes y = 2x+ 6.
Equations of Parallel and Perpendicular Lines
Understanding equations of parallel and perpendicular lines is similar to understanding equations from tables because it only requires an understanding of the points and slopes of each line. These can always be determined without ever looking at a graph.
If the equation for a line parallel to y = 2x + 1 is sought and it is known that this line passes through the point (3, 4), then the slope intercept form y = mx + b of a linear equation can be used to find the slope (m) of that parallel line. With the points x = 3 and y = 4 this means that 4 = 2(3) + b and the y-intercept of the new parallel line is b = –2. This means the new parallel line equation is y = 2x – 2.
If the equation for a line perpendicular to y = x + 4 is sought and it is known that this line passes through the point (−1, 3), then the slope intercept form y = mx + b of a linear equation can be used to find the slope (m) of that perpendicular line. The rule to remember here is that the slope of a perpendicular line is always the negative reciprocal of the slope of the original line. Since the original slope (m) = 1, the new slope is –1. Then the y-intercept can be found with the two points (–1, 3) substituted into the formula so that the result is 3 = –1(–1) + b and b = 2. The formula for the new perpendicular line is y = –x + 2.
Constructing and Solving Equations in the Real World
Equations can be used in the real world to determine many things. For example, it is possible to construct a linear equation where the solution is the number of apps that a person has on their smartphone. Rather than counting each of them, one by one, the apps could be multiplied according to all the complete rows and all the complete columns of icons on the phone and the remaining icons that come from incomplete rows or incomplete columns of icons added to the product. An equation is produced (a = bc + d) where a represents the total number of apps.
Bibliographies
Aufmann, Richard, Vernon Barker, and Richard Nation. College Algebra and Trigonometry. 7th ed. Belmont, CA: Cengage, 2011.
McKellar, Danica. Hot X: Algebra Exposed! New York: Plume, 2011.
Pickover, Clifford. The Math Book. New York: Sterling, 2012.
Polya, G. How to Solve It. Princeton: Princeton UP, 2014.
Strogatz, Steven. The Joy of X. New York: Mariner, 2013.
Young, Cynthia Y. Algebra and Trigonometry. 3rd ed. Hoboken, NJ: Wiley, 2013.