Math Introduction to Linear Equations

Math Introduction to Linear Equations

Math Introduction to Linear Equations 150 150 Deborah

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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