chapter 3 Algebra

"legal" moves

("legal" moves) When working with an equation mat or expression comparison mat, there are certain "legal" moves you can make with the algebra tiles that keep the relationship between the two sides of the mat intact. For example, removing an x tile from the positive region of each side of an equation mat is a legal move; it keeps the expressions on each side of the mat equal. The legal moves are those justified by the properties of the real numbers.

solve

(1) To find all the solutions to an equation or an inequality (or a system of equations or inequalities). For example, solving the equation x2 = 9 gives the solutions x = 3 and x = −3. (2) Solving an equation for a variable gives an equivalent equation that expresses that variable in terms of other variables and constants. For example, solving 2y − 8x = 16 for y gives y = 4x + 8. The equation y = 4x + 8 has the same solutions as 2y − 8x = 16, but y = 4x + 8 expresses y in terms of x and some constants.

standard form

1. The standard form for a linear equation is ax + by = c, where a, b, and c are real numbers and a and b are not both zero. For example, the equation 2.5x − 3y = 12 is in standard form. When you are given the equation of a line in standard form, it is often useful to write an equivalent equation in y = mx + b form to find the line's slope and y-intercept. 2. A quadratic expression in the form ax2; + bx + c is said to be in standard form. For example, the following are all expressions in standard form: 3m2 + m − 1, x2 − 9, and 3x2 + 5x.

equation

A mathematical sentence in which two expressions appear on either side of an "equals" sign (=), stating that the two expressions are equivalent. For example, the equation 7x + 4.2 = −8 states that the expression 7x + 4.2 has the value -8. In this course, an equation is often used to represent a rule relating two quantities. For example, a rule for finding the area y of a tile pattern with figure number x might be written y = 4x − 3.

closed sets

A set of numbers is said to be closed under an operation if the result of applying the operation to any two numbers in the set produces a number in the set. For example, the whole numbers are a closed set under addition, because if you add any two whole numbers the result is always a whole number. However, the whole numbers are not closed under division: if you divide any two whole numbers you do not always get a whole number.

generic rectangles

A type of diagram used to visualize multiplying expressions without algebra tiles. Each expression to be multiplied forms a side length of the rectangle, and the product is the sum of the areas of the sections of the rectangle. For example, the generic rectangle below can be used to multiply (2x + 5) by (x + 3).

algebra tiles

An algebra tile is a manipulative whose area represents a constant or variable quantity. The algebra tiles used in this course consist of large squares with dimensions x-by-x and y-by-y; rectangles with dimensions x-by-1, y-by-1, and x-by-y; and small squares with dimensions 1-by-1. These tiles are named by their areas: x2, y2, x, y, xy, and 1, respectively. The smallest squares are called "unit tiles." In this text, shaded tiles will represent positive quantities while unshaded tiles will represent negative quantities.

expression

An expression is a combination of individual terms separated by plus or minus signs. Numerical expressions combine numbers and operation symbols; algebraic (variable) expressions include variables. For example, 4 + (5 − 3) is a numerical expression. In an algebraic expression, if each of the following terms, 6xy2, 24, and are combined, the result may be 6xy2 + 24 − . An expression does not have an "equals" sign.

binomial

An expression that is the sum or difference of exactly two terms, each of which is a monomial. For example, −2x + 3y2 is a binomial.

polynomial

An expression that is the sum or difference of two or more monomials (terms). For example, x8 - 4x6y + 6x4y2 is a polynomial.

area

For this course, area is the number of square units needed to fill up a region on a flat surface. In later courses, the idea will be extended to cones, spheres, and more complex surfaces.

terms

For this course, area is the number of square units needed to fill up a region on a flat surface. In later courses, the idea will be extended to cones, spheres, and more complex surfaces.

exponent

In an expression of the form ba, a is called the exponent. For example, in the expression 25, 5 is called the exponent. (2 is the base, and 32 is the value.) The exponent indicates how many times to use the base as a multiplier. For example, in 25, 2 is used 5 times: 25 = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 = 32. For exponents of zero, the rule is: for any number x ≠ 0, x0 = 1. For negative exponents, the rule is: for any number x ≠ 0, , and . (Also see laws of exponents.)

dimensions

The dimensions of a flat region or space tell how far it extends in each direction. For example, the dimensions of a rectangle might be 16 cm wide by 7 cm high.

solution

The number or numbers that when substituted into an equation or inequality make the equation or inequality true. For example, x = 4 is a solution to the equation 3x − 2 = 10 because 3x − 2 equals 10 when x = 4. A solution to a two-variable equation is sometimes written as an ordered pair (x, y). For example, x = 3 and y = −2 is a solution to the equation y = x − 5; this solution can be written as (3, −2).

product

The result of multiplying. For example, the product of 4 and 5 is 20; the product of 3a and 8b2 is 24ab2.

integers

The set of numbers { . . . −3, −2, −1, 0, 1, 2, 3, . . . }

evaluate

To evaluate an expression, substitute the value(s) given for the variable(s) and perform the operations according to the order of operations. For example, evaluating 2x + y − 10 when x = 4 and y = 3 gives the value 1.

Distributive Property

We use the Distributive Property to write a product of expressions as a sum of terms. The Distributive Property states that for any numbers or expressions a, b, and c, a(b + c) = ab + ac. For example, 2(x + 4) = 2 · x + 2 · 4 = 2x + 8. We can demonstrate this with algebra tiles or in a generic rectangle.

base

When working with an exponential expression in the form ba, b is called the base. For example, 2 is the base in 25 (5 is the exponent, and 32 is the value.) Also see exponent.

sum

answer to a problem