Overview:
Linear equations are prevalent in algebra when the relationship between variables and constants can be plotted on a straight line. For example, suppose the equation were 2x = y. If the value for x were 1, the value for y would be 2, or for x were 2, y would be 4, and so forth.
What Are Linear Equations?
Suppose the equation were 2x – 1 = 7. There is only one value of x that would make the statement true, if x equaled 4, because 2(4) – 1 equals 7. However, if the equation were 2x – y = 7, there would be many values for x and for y, such that if x equaled 4, then y would equal 1, if x equaled 5, y would equal 2, and so on. Although only 2 points are needed to plot a straight line, it is useful to have a third as a check. If x equaled 2, y would equal -3, because 4 – (-3) would equal 7.
What Is the Slope of a Line?
The slope of a line is the ratio between the vertical change to the horizontal change, as long as the horizontal change is not equal to 0. Suppose the equation were 2x = y. If x equals 1, y equals 2, if x equals 2, y equals 4 and so forth. The vertical change is 2 and the horizontal change is 1, so the slope is 2/1.
Where Does the Line Cross the Y-Axis?
Suppose the equation were 3x – y = 11. If x were 4, y would be 1, so its coordinates would be (4, 1), and if x were 2, y would be -5, so its coordinates would be (2, -5). What if x were equal to 0? If x were 0, y would be -11, and its coordinates would be (0,-11). The point at which the line crosses the y axis is the point at which x equals 0. It is called the y-intercept.
What Is the Slope-Intercept Form of a Line?
The slope-intercept form of a line is the equation for the line that contains both the slope and the y-intercept. Suppose the equation were 2x + y = 5. In order to solve for y the equation would be y = -2x + 5. If y were equal to 5, x would equal 0 (the y intercept), and its coordinates would be (0, 5). If y were equal to 1, x would equal 2, and its coordinates would be (2, 1). The slope of the line would be the ratio of the vertical change to the horizontal change or (5-1)/(0-2) or 4/-2 or -2.
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