Segments, Lines and Inequalities
Equations of Lines
Equations involving one or two variables can be graphed on any x-y coordinate plane. In general, it is TRUE that: ➜ If a point lies on the graph of an equation, then its coordinates make the equation a true statement, AND ➜ If the coordinates of a point make an equation a true statement, then the point lies on the graph of the equation. ➜ The GRAPHS OF LINEAR EQUATIONS ARE ALWAYS LINES. ➜ All linear equations can be written in the form Ax + By = C, where A, B, and C are real numbers and A and B are not both zero. ➜ Furthermore, to be in standard form, A has to be a positive number.
Ordered Pairs
Every point in a coordinate plane is named by a PAIR OF NUMBERS WHOSE ORDER IS IMPORTANT. This pair of numbers, written in parentheses and separated by a comma, is the ordered pair for the point
Graphing Linear Equations
One way to graph a linear equation is to find solutions by giving a value to one variable and solving the resulting equation for the other variable. A minimum of two points is necessary to graph a linear equation. Copy and paste the following link into your browser to learn more about graphing a linear In https://youtu.be/5h6YzRRxzO4
Parallel Lines
PARALLEL LINES are lines in the same plane that do not intersect.
Point-Slope Form of a Non-Vertical Line
Point-Slope Form of a Line is the equation of the non-vertical line passing through the points (x₁,y₁) and (x₂, y₂) and having slope m is given by the equation: y - y₁ = m ( x - x₁ ). Which point you call point 1 and which point you call point 2 does not matter. Copy and paste the following link into your browser to learn more about using the point-slope form: http://www.bing.com/videos/search?q=point+slope+form+of+a+nonvertical+line&&view=detail&mid=8C1285373C18BC065B918C1285373C18BC065B91&FORM=VRDGAR
Slope of Parallel and Perpendicular Lines
SLOPE OF PARALLEL LINES: Parallel lines have equal slopes. Stated another way, if two different lines have equal slopes, then they are parallel lines. SLOPE OF PERPENDICULAR LINES If two nonvertical lines are perpendicular, then their slopes are negative reciprocals (actually, opposite reciprocals) of one another, or the product of their slopes is -1. Stated another way, if the slopes of two lines are opposite reciprocals of one another, or the product of their slopes is -1, then the lines are nonvertical perpendicular lines. Because horizontal and vertical lines are always perpendicular, then lines having a zero slope and an undefined slope are perpendicular.
Rectangular Coordinate System
The RECTANGULAR COORDINATE SYSTEM is also called as the Cartesian coordinate system or the X-Y coordinate system. The rectangular coordinate system is composed of a set of two intersecting lines. The horizontal axis is commonly labeled as the X-axis and the vertical axis as the Y-axis. The X and Y axes divide the system into four parts that are called the quadrants. Any point in the rectangular coordinate system is corresponding to an ordered pair (x, y). This pair notation is called ordered pair. Here, x and y are real numbers. The number x is the X coordinate or horizontal axis or abscissa and the number y is the Y coordinate or vertical axis or ordinate. The origin is a point where these two axes cross.
Slope of a Line and Slope Formula
The SLOPE OF A LINE is a measurement of the steepness and direction of a nonvertical line. When a line slants from lower left to upper right, the slope is a POSITIVE number. When a line slants from upper left to lower right, the slope is a NEGATIVE number. The x‐axis or any line parallel to the x‐axis has a slope of zero; that is, a horizontal line has a slope of ZERO. The y‐axis or any line parallel to the y‐axis has no defined slope; that is, a vertical line has an undefined slope. Copy and paste the following link into your browser to learn more about using the slope formula to find the slope of a line [positive, negative or zero]: http://www.bing.com/videos/search?q=using+the+slope+formula+to+find+the+slope+of+a+line&qpvt=using+the+slope+formula+to+find+the+slope+of+a+line&view=detail&mid=92DC66C2B226DA9ECBE692DC66C2B226DA9ECBE6&FORM=VRDGAR
X-Intercept
The X-INTERCEPT of a graph is the point at which the graph will intersect the x‐axis. It will always have a y‐coordinate of zero. A horizontal line that is not the x‐axis will have no x‐intercept.
Y-Intercept
The Y‐INTERCEPT of a graph is the point at which the graph will intersect the y‐axis. It always has an x‐coordinate of zero. A vertical line that is not the y‐axis has no y‐intercept.
Distance Formula
The distance formula is derived from the Pythagorean theorem. To find the distance between two points (x₁, y₁) and (x₂, y₂), all that you need to do is use the coordinates of these ordered pairs and apply the formula pictured below. The distance formula is Distance = √ (x₂ − x₁)² +(y₂−y₁)² Copy and paste the following link into your browser to learn more about using the distance formula to calculate the distance between two points (x₁, y₁) and (x₂, y₂): https://youtu.be/PuqdjXyBavY
Standard Form
The form 'Ax + By = C' for the equation of a line is known as the STANDARD FORM for the equation of a line.
X-Coordinate
The number to the left of the comma in an ordered pair is the X‐COORDINATE of the point and indicates the amount of movement parallel to the x‐axis from the origin. The movement is to the right if the number is positive and to the left if the number is negative.
Y-Coordinate
The number to the right of the comma in an ordered pair is the Y-COORDINATE of the point and indicates the amount of movement perpendicular to the x‐axis. The movement is above the x‐axis if the number is positive and below the x‐axis if the number is negative.
Graph of the Ordered Pair
The point associated with an ordered pair of real numbers is called the GRAPH OF THE ORDERED PAIR.
Origin
The point of intersection of the x‐axis and the y‐axis is called the ORIGIN. The ordered pair for the origin is (0,0).
Quadrants
The x‐axis and y‐axis separate the coordinate plane into four regions called QUADRANTS. The upper right quadrant is quadrant I, the upper left quadrant is quadrant II, the lower left quadrant is quadrant III, and the lower right quadrant is quadrant IV. ➜ IN QUADRANT # 1: x is always positive and y is always positive (+,+) ➜ IN QUADRANT # 2: x is always negative and y is always positive (-,+) ➜ IN QUADRANT # 3: x is always negative and y is always negative (-,-) ➜ IN QUADRANT # 4: x is always positive and y is always negative (+,-)
Coordinate Plane
The x‐axis, the y‐axis, and all the points in their plane are called a COORDINATE PLANE.
X-Axis and Y-Axis
To locate points in a plane, two perpendicular lines are used: a horizontal line called the X-AXIS and a vertical line called the Y-AXIS.
Perpendicular Lines
Two lines that intersect to form right angles are called PERPENDICULAR LINES.
Coordinates of a Point
COORDINATES OF A POINT: Each point on a number line is assigned a number. In the same way, each point in a plane is assigned a pair of numbers called the coordinates of the point.
Linear Inequalities
LINEAR INEQUALITY is a sentence in one of the following forms: ☛ Ax + By < C ☛ Ax + By > C ☛ Ax + By ≤ C ☛ Ax + By ≥ C To graph such sentences: ➜ [1] Graph the linear equation Ax + By = C. This line becomes a boundary line for the graph. If the original inequality is < or >, the boundary line is drawn as a dashed line, since the points on the line do not make the original sentence true. If the original inequality is ≤ or ≥, the boundary line is drawn as a solid line, since the points on the line will make the original inequality true. ➜ [2] Select a point not on the boundary line and substitute its x and y values into the original inequality. ➜ [3] Shade the appropriate area. If the resulting sentence is true, then shade the region where that test point is located, indicating that all the points on that side of the boundary line will make the original sentence true. If the resulting sentence is false, then shade the region on the side of the boundary line opposite that where the test point is located. Copy and paste the following link into your browser to learn more about using graphing linear inequalities: https://youtu.be/P_-c9D6mjGA
Midpoint Formula
Numerically, the midpoint of a segment can be considered to be the average of its endpoints. This concept should help in remembering a formula for finding the midpoint of a segment, given the coordinates of its endpoints. Recall that the average of two numbers is found by dividing their sum by two Copy and paste the following link into your browser to learn more about using the midpoint formula: http://www.bing.com/videos/search?q=using+the+midpoint+formula&&view=detail&mid=04668C465484A26C135304668C465484A26C1353&FORM=VRDGAR