Linear Equations and Inequalities: Cheat Sheet

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Essential Concepts

Graphing and Analyzing Linear Equations

  • We can locate or plot points in the Cartesian coordinate system using ordered pairs which are defined as displacement from the x-axis and displacement from the y-axis.
  • An equation can be graphed in the plane by creating a table of values and plotting points.
  • The slope of a line is a measure of its steepness or the angle at which it tilts, expressed as the ratio of the rise (the vertical change) to the run (the horizontal change) between any two points on the line. It quantifies how much the line goes up or down as it moves from left to right.
    • A positive slope means that the line rises from left to right.
    • A negative slope means that the line falls from left to right.
    • A slope of zero means the line is flat.
  • Finding the [latex]x[/latex]– and [latex]y[/latex]–intercepts can define the graph of a line. These are the points where the graph crosses the axes.
    • The [latex]x[/latex]-intercept is the point at which the graph crosses the [latex]x[/latex]-axis. At this point, the [latex]y[/latex]-coordinate is zero.
    • The [latex]y[/latex]-intercept is the point at which the graph crosses the [latex]y[/latex]-axis. At this point, the [latex]x[/latex]-coordinate is zero.
  • The distance formula is derived from the Pythagorean Theorem and is used to find the length of a line segment.
  • The midpoint formula provides a method of finding the coordinates of the midpoint by dividing the sum of the [latex]x[/latex]-coordinates and the sum of the [latex]y[/latex]-coordinates of the endpoints by [latex]2[/latex].

Equations of Lines

  • To evaluate an algebraic expression, you substitute the given values for the variables in the expression. Then you simplify the expression using the order of operations, just like you would with regular math problems. If there is more than one variable, you replace each variable with its assigned value and simplify the expression in the same way.
  • To solve a multi-step equation, follow these steps:
    • (Optional) If there are fractions or decimals, you can multiply the whole equation by a number to get rid of them.
    • Simplify both sides of the equation by removing parentheses and combining similar terms.
    • Add or subtract terms to isolate the variable term. You might need to move terms around to achieve this.
    • Multiply or divide to get the variable by itself.
    • Finally, check if the solution you found makes the equation true.
  • The slope-intercept form of a line is written as: [latex]y=mx+b[/latex], where [latex]m[/latex] is the slope and [latex]b[/latex] is the y-intercept.
  • The point-slope form of the equation of a line is: [latex]y-y_1 = m(x-x_1)[/latex], where [latex]m[/latex] is the slope and [latex](x_1, y_1)[/latex] is the coordinate of any point on the line.
  • The standard form of a line is: [latex]Ax+By = C[/latex], where [latex]A, B, C[/latex] are integers and [latex]A[/latex] and [latex]B[/latex] are not zeros.

Modeling with Linear Equations

  • A linear equation can be used to solve for an unknown in a number problem.
  • Applications can be written as mathematical problems by identifying known quantities and assigning a variable to unknown quantities.
  • There are many known formulas that can be used to solve applications. Distance problems, for example, are solved using the [latex]d=rt[/latex] formula.
  • Many geometry problems are solved using the perimeter formula [latex]P=2L+2W[/latex], the area formula [latex]A=LW[/latex], or the volume formula [latex]V=LWH[/latex].

Linear Inequalities

  • Interval notation is a method to give the solution set of an inequality. Highly applicable in calculus, it is a system of parentheses and brackets that indicate what numbers are included in a set and whether the endpoints are included as well.
  • Solving inequalities is similar to solving equations. The same algebraic rules apply, except for one: multiplying or dividing by a negative number reverses the inequality.
  • Compound inequalities often have three parts and can be rewritten as two independent inequalities. Solutions are given by boundary values which are indicated as a beginning boundary or an ending boundary in the solutions to the two inequalities.

Key Equations

Slope-Intercept Form of a Line [latex]y = mx + b[/latex]
Point-Slope Form of a Line [latex]y - y_1 = m(x - x_1)[/latex]
Standard Form of a Line [latex]Ax + By = C[/latex]
Distance Formula [latex]d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/latex]
Midpoint Formula [latex]( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} )[/latex]
Perimeter of a Rectangle [latex]P = 2L + 2W[/latex]
Area of a Rectangle [latex]A = LW[/latex]

Glossary

algebraic expression
a mathematical phrase or combination of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division
area
in square units, the area formula used in this section is used to find the area of any two-dimensional rectangular region: [latex]A=LW[/latex]

Cartesian coordinate system

a grid system designed with perpendicular axes invented by René Descartes

compound inequality
a problem or a statement that includes two inequalities
distance formula
a formula that can be used to find the length of a line segment if the endpoints are known
equation in two variables
a mathematical statement, typically written in x and y, in which two expressions are equal
graph in two variables
the graph of an equation in two variables, which is always shown in two variables in the two-dimensional plane
intercepts
the points at which the graph of an equation crosses the x-axis and the y-axis
interval
an interval describes a set of numbers where a solution falls
interval notation
a mathematical statement that describes a solution set and uses parentheses or brackets to indicate where an interval begins and ends
linear inequality
similar to a linear equation except that the solutions will include an interval of numbers
midpoint formula
 a formula to find the point that divides a line segment into two parts of equal length
ordered pair
a pair of numbers indicating horizontal displacement and vertical displacement from the origin; also known as a coordinate pair, [latex]\left(x,y\right)[/latex]
origin
the point where the two axes cross in the center of the plane, described by the ordered pair [latex]\left(0,0\right)[/latex]
perimeter
in linear units, the perimeter formula is used to find the linear measurement, or outside length and width, around a two-dimensional regular object; for a rectangle: [latex]P=2L+2W[/latex]
quadrant
one quarter of the coordinate plane, created when the axes divide the plane into four sections
slope
the change in [latex]y[/latex]–values over the change in [latex]x[/latex]–values

variable

a symbol that represents a value or quantity that can change or vary in a given situation or context

volume
in cubic units, the volume measurement includes length, width, and depth: [latex]V=LWH[/latex]
x-axis
the common name of the horizontal axis on a coordinate plane; a number line increasing from left to right
x-coordinate
the first coordinate of an ordered pair, representing the horizontal displacement and direction from the origin
x-intercept
the point where a graph intersects the x-axis; an ordered pair with a y-coordinate of zero
y-axis
the common name of the vertical axis on a coordinate plane; a number line increasing from bottom to top
y-coordinate
 the second coordinate of an ordered pair, representing the vertical displacement and direction from the origin
y-intercept
a point where a graph intercepts the y-axis; an ordered pair with an x-coordinate of zero