What happens if we multiply both numbers by the same value c? The exercise below will let us find out. On this number line, points B and A are our original values of 2 and 5. The inequality has been maintained.
Systems of Equations and Inequalities In previous chapters we solved equations with one unknown or variable. We will now study methods of solving systems of equations consisting of two equations and two variables.
Represent the Cartesian coordinate system and identify the origin and axes. Given an ordered pair, locate that point on the Cartesian coordinate system. Given a point on the Cartesian coordinate system, state the ordered pair associated with it.
We have already used the number line on which we have represented numbers as points on a line.
Note that this concept contains elements from two fields of mathematics, the line from geometry and the numbers from algebra. Rene Descartes devised a method of relating points on a plane to algebraic numbers.
This scheme is called the Cartesian coordinate system for Descartes and is sometimes referred to as the rectangular coordinate system.
This system is composed of two number lines that are perpendicular at their zero points. Perpendicular means that two lines are at right angles to each other.
Study the diagram carefully as you note each of the following facts. The number lines are called axes. The horizontal line is the x-axis and the vertical is the y-axis. The zero point at which they are perpendicular is called the origin.
Positive is to the right and up; negative is to the left and down. The arrows indicate the number lines extend indefinitely. Thus the plane extends indefinitely in all directions.
The plane is divided into four parts called quadrants. These are numbered in a counterclockwise direction starting at the upper right. Points on the plane are designated by ordered pairs of numbers written in parentheses with a comma between them, such as 5,7.
This is called an ordered pair because the order in which the numbers are written is important. The ordered pair 5,7 is not the same as the ordered pair 7,5. Points are located on the plane in the following manner. First, start at the origin and count left or right the number of spaces designated by the first number of the ordered pair.
Second, from the point on the x-axis given by the first number count up or down the number of spaces designated by the second number of the ordered pair. Ordered pairs are always written with x first and then y, x,y. The numbers represented by x and y are called the coordinates of the point x,y.
The first number of the ordered pair always refers to the horizontal direction and the second number always refers to the vertical direction.
Check each one to determine how they are located.There are no ordered pairs that satisfy both equations. - Consistent equations can be independent or dependent.
If a system has exactly one solution, it is independent. If the system has an infinite number or solutions, it is dependent 1.) Number of Solutions 2.) Solve a System of . The solution region for the previous example is called a "closed" or "bounded" solution, because there are lines on all sides.
That is, the solution region is a bounded geometric figure (a triangle, in that case). Since the two shaded areas don’t intersect or overlap, this tells us that the given system of inequalities has NO SOLUTION. You may have also observed that the boundary lines have equal slopes, both m = –2, which implies that they are parallel lines, and therefore won’t intersect.
A system of linear inequalities in two variables consists of at least two linear inequalities in the same variables. The solution of a linear inequality is the ordered pair that is a solution to all inequalities in the system and the graph of the linear inequality is the graph of all solutions of the system.
The important thing about inequalities is that there are multiple possible solutions.
For example, the inequality “50 ≥ the number of stars on any United States flag” is a true statement for every American flag ever flown—no flag has had over 50 stars. Sep 12, · Best Answer: Two inequalities have no solution when both of them must be true and they result in mutually exclusive conditions.
For example: x + 5 > 10 and x -2 5 and x no number that is both greater than 5 and less than 3, therefore there is no alphabetnyc.com: Resolved.