Pre Algebra Linear Equations in Two Variables

Linear Equations in Two Variables The Coordinate Plane Two perpendicular number lines separate a plane into four regions called quadrants. The quadrants are numbered with Roman numerals, in the counterclockwise direction. ● The x-axis is the horizontal number line, and the y-axis is the vertical number line. ● The origin is the intersection between the x and y axes, represented by the ordered pair (0,0). ● Points being located on a plane are denoted using ordered pairs (x,y). The first number in an ordered pair represents the x-coordinate. The second is the y-coordinate. ○ A plane that has been set up in this way is referred to as a coordinate plane. ● When you’re graphing on the coordinate plane, first find the x-coordinate, then the ycoordinate. These directions start at the origin. Positive Negative x-axis right left y-axis up down ● When graphing an equation, you are creating a picture/graph of the ordered pairs that are solutions to the equations (using the rule). Graphs show you how many solutions there are to the equation. Some graphs form lines when multiple solutions are plotted. ● The graph of the line represents a linear relationship between the two variables, x and y. ● Steps to draw a graph: 1. Create a table of values to represent the relationship or equation. 2. Graph ordered pairs (from table) 3. Determine if relationship is proportional or non-proportional Slope as a Rate of Change- From Graphs ● In a linear relationship, the rate of change on a graph is the slope of the line. This term is used to describe the steepness of a line. It is the ratio of rise over run, (vertical change over horizontal change). ● We can measure the steepness of a line comparing how much the y-variable changes over the x-variable change ● Slope (rate of change) can be positive or negative. ○ Positive: line slants upward left to right ○ Negative: line slants downward from left to right ○ Horizontal lines have a slope of zero. ○ Vertical lines have an undefined slope (the slope cannot be determined). Sketching Graphs of Linear Equations in y=mx+b ● This is called slope intercept form. ● M represents the slope and the b represents the y-intercept of the line. ● A linear equation has an infinite number of solutions. The solutions are all the ordered pairs that make the equation true. ○ You can create a table of values and graph the ordered pairs to determine the line. (m= rise/run) Rewriting Equations in Slope Intercept Form ● It is easy to graph an equation written in slope-intercept form. In order to graph an equation not written in slope intercept form, isolate the y on one side of the equation by using the properties of equality. Slope as a Rate of Change- From Two Points ● To find slope from 2 points, use the Slope Formula: ○ The slope of a line passing through points (x1, y1) and (x2, y2) is the ratio of the difference in the y-coordinates to the difference in the x-coordinates. Rate of Change from a Table ● The slope is a rate that describes how a quantity is in relation to another quantity. In a linear relationship, the rate of change between any two ordered pairs is constant. This is called the constant rate of change. ● You can determine if a relationship is linear from a table of values, without graphing the line, by determining if there is a constant rate of change between the ordered pairs in the table. Linear Equations in Slope- Intercept Form ● Linear relationships can be written in the form y=mx+b (slope intercept form). ○ M, the coefficient of x, represents the slope. ○ The constant, b, is the y-value of the y-intercept (0, b). ○ When an equation in slope-intercept form applies to a real-world situation, the slope represents the rate of change, and the y-intercept represents the initial value.