Part 2: Linear Equations
Linear equations can be used to model situations involving one or more unknowns. A linear equation is an algebraic equation whose variable(s) is/are of degree one. For example, 6x + 3 = 27 and a = 2b are linear equations. Solving linear equations is a key component of the algebra curriculum.
A linear equation is a polynomial equation of the first degree, such as x + y = 7. Said another way, a linear equation has no variables raised to a power other than one.
The simplest linear equations involve only one unknown, such as x + 2 = 3, and they are solved by finding the value of the unknown that makes the equation true. For example, the equation above is true when x = 1, because 1 + 2 = 3. More complex linear equations may contain more than one variable, such as x + y = 7 or a - b + c - d + 4 = 14. This workshop will focus on the simplest linear equations and those whose graph is a line.
Linear equations with just one unknown are solved using equivalence transformations, sometimes informally called "inverse operations." In this process, the operation of multiplication is "undone" using division, because multiplication and division are multiplicative inverses; similarly, addition is "undone" using subtraction, because addition and subtraction are additive inverses. The reason this process works is because the equal sign acts as a balance between the left side and the right side of the equation. As long as the same operation is applied to each side of the equation, the equation remains in balance and the equality is preserved. For example, the equation 2x + 3 = 9 is solved as follows:
Subtraction is the additive inverse of addition. Therefore, to "undo" the addition of 3, subtract 3 on both sides of the equation:
2x + 3 - 3 = 9 - 3, which yields 2x = 6.
Because division is the multiplicative inverse of multiplication, to "undo" the multiplication of x by 2, divide both sides by 2:
, which yields x = 3.
- A traditional example involves people's ages. For instance, "Becky is 6 years younger than Sally, and Sally is 13 years old. What is Becky's age?" The equation b + 6 = 13 can represent this situation, and it is true when b = 7, so Becky is 7 years old.
- A person bought 3 cans of soda as well as several six-packs of soda, and she has a total of 27 cans. The linear equation 6x + 3 = 27 determines the number of six-packs she bought, where x is the number of six-packs.
- Linear functions can model the "transmission factor" of two gears. The ratio between the number of revolutions made by a 1 cm gear (b) and the number of revolutions made by a 2 cm gear (a) can be expressed as a = 0.5b and b = 2a.
Linear Equation or Expression: An algebraic equation or expression which is of the first degree in its variable (or variables); i.e., its highest degree term in the variable (or variables) is of the first degree. The equations x + 2 = 0 and x + y + 3 = 0 are linear. An equation or expression is linear in a certain variable if it is of the first degree in that variable. The equation x + y2 = 0 is linear in x, but not in y.
(Source: James, Robert C. and Glenn James, Mathematics Dictionary (4th edition). New York: Chapman and Hall, 1976.)
John McLeish provides a useful explanation of linear: "Linear means an equation of the first power of the unknown; such an equation can be represented by a straight line graph, hence 'linear.'" (McLeish, John. Number: The History of Numbers and How They Shape Our Lives. New York: Fawcett Columbine, 1991.)
Role in the Curriculum
To become comfortable solving and manipulating linear equations, students will need to experience linear relationships in various situations. They will need a significant amount of practice before developing fluency.
Upon the successful completion of an algebra course, students should be able to use symbolic notation to represent and explain mathematical relationships and solve linear equations. The National Council of Teacher of Mathematics (NCTM) states:
Although students will probably acquire facility with equations at different times ...
students should be able to solve equations like 84 - 2x = 5x + 12 for the unknown number ... and to recognize that equations such as y = 3x + 10 represent linear functions that are satisfied by many ordered pairs (x, y).
Students should also be able to produce two or more equivalent expressions that represent the same situation and to use simple formulas.
(Principles and Standards for School Mathematics, NCTM, 2000, p. 226)
Solving linear equations is a key part of attaining a global understanding of linear relationships and involves more than the ability to solve for an unknown. The solution of linear equations is often a necessary step when interpreting a complex situation. Students may describe a linear relationship using a table, graph, or words, and from that description they may generate an expression or equation to represent the situation. For instance, students may describe the cost of a cell phone plan in various ways:
Some students might describe the pattern verbally: "Keep-in-Touch costs $20.00 [monthly] and then $0.10 more per minute [of use.]" Others might write an equation to represent the cost (y) in dollars in terms of the number of minutes (x), such as y = 20.00 + 0.10x.
From the function, students should be able to find the cost for any number of minutes. The cost for 25 minutes occurs when x = 25, and finding the associated cost involves solving the equation y = 20.00 + 0.10(25). Similarly, the number of minutes for which the cost would be $35.00 occurs when y = 35.00, and students can find that value by solving the equation 35.00 = 20.00 + 0.10x.
(PSSM, p. 226)