Algebra

Variable vs Constant

A variable, often designated with a letter (x, y, or z, for example) is used to represent an unknown quantity that may change its value over the course of a problem.

A constant, on the other hand, is a value that does not change, regardless of the circumstances.

Consider the following situation: John is paid a weekly base wage of $15. In addition to this base wage, he earns $0.25 for every magazine he sells. Write an equation representing John’s total wage.

In this case, the base wage John earns is a constant, so regardless of the number of magazines he sells, he will gain $15. If he sells 0 magazines, he will earn $15. If he sells 1 magazine, he will earn $15 + $0.25, or $15.25 . We can use a variable to represent the unknown number of magazines he sells and combine this with his base wage to express his total wage. Let m represent the unknown number of magazines sold, and let f(m) be his total wage:

f(m) = 0.25 * m + 15

Because he earns $0.25 per magazine sold, when we find the product of this with the number of magazines sold, m, we are finding the total amount of money earned from the magazines he has sold. By combining this with his base wage, we calculate his total wage.

Coordinate Plane

The coordinate plane provides a two-dimensional space with which to graphically represent points, lines, and functions in two or fewer variables; it is also known as the xy-axis. The horizontal axis is the x-axis, and the vertical axis is the y-axis. The origin is the intersection of the x-axis and the y-axis and is located at point (0, 0). This intersection creates four quadrants.

These four quadrants begin with the top right as quadrant I. The quadrants increase in number in a counterclockwise fashion. The quadrant to the left of quadrant I is designated quadrant II. The quadrant below quadrant II is quadrant III, and the remaining quadrant is quadrant IV.

• Quadrant I contains positive x and positive y values.

• Quadrant II contains negative x and positive y values.

• Quadrant III contains negative x and negative y values.

• Quadrant IV contains positive x and negative y values.

The coordinate plane is most useful for showing the relationship between two variables. The x variable is commonly referred to as the independent variable (meaning the value of the variable is chosen), and the y variable is commonly referred to as the dependent variable (meaning its value is determined by the function containing the independent variable).

Points, given in the form (x, y), are commonly graphed on the coordinate plane. The point (-3, 2), for example, would be located three units to the left of the origin (0, 0), and two units above the x-axis.

Functions of the form f(x) = can be graphed in the coordinate plane by selecting values for x, determining their corresponding y values, and then placing points at each of the calculated (x, y) values. The nature of the function determines whether a straight or curved line can be drawn to connect each of the points.

Slope and Intercept

The slope of a line is the ratio of the change in y values to the change in x values:

m = (y2 - y1)/(x2 - x1)

where

(x1, y1) and (x2, y2)

are points on the line.

It is helpful to think of the slope as the rise over the run, but it is also important to understand that a line is straight because the change in the y values is in proportion with the change in the x values. If this was not so, the graph would exhibit a different characteristic.

Aside from the slope, graphs of lines will also exhibit x or y-intercepts. These are points at which lines cross the x or y axis. These points are important because they show the x value of the function when y is equal to 0, and the y value of the function when x is equal to 0.

Linear Equation vs Linear Expression

The fundamental difference between an equation and an expression is that an equation contains two expressions that are equal to each other. This equality is represented with the equal sign, =.

Expressions cannot be solved. Instead, where possible, they can be simplified, or reduced. Consider the following expression:

4x − 2x + 3x2 + 5x2

The terms containing the same variable type can be combined and the expression can be simplified:

4x − 2x = 2x

and

3x2 + 5x2 = 8x2

so the expression can be rewritten as:

2x + 8x2

Remember that only like terms can be combined. 2x and 2x2 cannot be combined because the variable is to a different degree.

A linear equation containing one variable can be numerically solved for the variable. A linear equation containing two or more variables can only be solved in terms of those variables. For example:

2y + 3x − z = 10

We can solve this equation for x, y or z, but the other side of the equation will still contain the other two variables. To solve for all three variables, we must have at least three distinct equations.

Linear Inequality

A linear inequality compares x and y expressions using >, ≥, <, or ≤. Linear inequalities are simplified in the same way that linear equations are simplified, with one major difference. When a linear inequality is multiplied or divided by a negative integer, the direction of the inequality sign switches, for example:

−4x < 8

x > −2

Rational Coefficient

Rational coefficient is a term for any number that is the ratio of two integers with the denominator of the ratio not equalling 0.

2, 0/1, 1/25, and square root of 25 are all rational coefficients.

30 and square root of 2 are not considered rational coefficients; the first is undefined, and the second cannot be represented as the fraction of two integers.

Linear Function

A linear function is a function of one variable that graphs as a line. The input value is commonly called x, and the output value f(x) is commonly called y.

Consider the following linear function:

f(x) = 3x − 2

To graph this function, input 2 distinct x values and find their corresponding f(x) values. Plot both points on the coordinate plane and draw a straight line extending through them.

Familiarity with the common equations for a line sometimes simplifies the graphing process. You should be comfortable using:

Slope-intercept form — y = mx + b

Point-slope form — (y − y1) = m(x − x1)

Standard form — Ax + By = C

Where m is the slope, b is the y-intercept, (x1, y1) is a point on the line, and A, B, and C are constants.

Solution Set

The solution set of an equation or inequality is the collection of values that yields a true statement when plugged into the original equation or inequality.

Consider:

4x = 8

In this case, the solution set is {2}.

Consider:

y >= 2x

In this case, the solution set is every point along the line and every point above the line.

Let’s confirm this by testing two points: (0, 0) and (3, −3). Substituting the first,

0 >= 2 * 0

0 >= 0 is true and (0, 0) is part of the solution set.

−3 >= 2 * 3

−3 >= 6 is not true and (3, −3) is not part of the solution set.

Constraints

One powerful tool of mathematics is its usefulness in helping us model real-world situations. Often, a function will only yield meaningful results when restrictions, or constraints, are placed upon it.

Constraints are often represented or established using inequalities. If, for example, only a positive collection of inputs and output values are necessary, this constraint can be accomplished by stating that both x and y must be greater than 0.

Regarding constraints, it is important to distinguish between the expressions at least and at most. The first can be translated mathematically as a less than or equal to inequality. The second expression can be translated as a greater than or equal to inequality.

Real-world situations involving time often require constraints because negative time is nonsensical.

No Solution vs Infinite Solutions

When solving a system of linear equations, three solution types are possible. No solution, one solution, and infinite solutions.

• No solution corresponds to the case where the two lines share the same slope (they are parallel), and they are not multiples of each other.

• One solution corresponds to the case where the lines intersect exactly once. This occurs in the case where the lines are neither parallel nor the same.

• In the case where the two lines are exactly the same, or integer multiples of each other, the system has an infinite number of solutions.

Absolute Value Expressions, Equations, and Inequalities

The absolute value of a number or expression is its distance from 0. Absolute values are always positive because they represent a magnitude. Expressions and equations containing absolute values (represented by an expression surrounded by two vertical lines) can be manipulated in much the same way expressions and equations lacking absolute values can be manipulated.

Consider the following:

−3|2x + 3| = −12

The goal, in this case, is to isolate the absolute value; this is accomplished by dividing both sides by −3:

|2x + 3| = 4

To solve an absolute value that has been isolated, rewrite the problem as two equations, one equal to positive 4 and one equal to −4. Solve each equation for x:

|2x + 3| = 4 is rewritten as:

2x + 3 = 4 and 2x + 3 = −4

So:

x = 1/2

x = −7/2

These answers can be verified by substituting them back into x and evaluating.

Inequalities can also contain an absolute value. In the case of inequalities containing less than or less than and equal to signs (<, ≤), the original inequality must be rewritten without the absolute value to indicate that the solution set includes all values between (and sometimes including) the positive and negative integer expressions in the original inequality. For example:

|6x − 8| ≤ 52

is rewritten as:

−52 ≤ 6x − 8 ≤ 52

Which can be broken apart to form two inequalities that can be solved individually:

−52 ≤ 6x − 8 and 6x − 8 ≤ 52

So,

−44/6 ≤ x and x ≤ 10

This solution set would be graphed as two closed (shaded) circles at x = −44/6 and x = 10, with the values between them being shaded as well. The closed circles indicate that the values at the beginning and end of the shaded line are also part of the solution.

Greater than or greater than or equal to inequalities containing an absolute value are solved in a similar fashion, for example:

2|3x − 2| ≥ 18

In this case, division by 2 results in:

|3x − 2| ≥ 9

Which can now be rewritten as:

3x− 2 ≥ 9

or

3x − 2 ≤ −9

So,

x ≥ 113

or

x ≤ −73

This solution set would be graphed with one closed circle at x = 11/3 and an arrow pointing to the right indicating that all values larger than 11/3 are also true. Another closed circle would be at x = −7/3 with an arrow pointing to the left of this value indicating that all numbers smaller than −7/3 are also true.