Thursday, July 16, 2020

Linear Equations Review - Notes

Here are the lecture notes for the Linear Equations Review Section of the Linear Programming Unit.  Let me know if you have any questions or problems with accessing the material.


Linear Equations Review. First, we discuss the Cartesian Coordinate System. The graph is displayed with the x-axis, the y-axis, the origin, and the point (a, b). The point (a, b) is graphed by moving a units to the right from the origin on the x-axis and then up b units from the origin on the y-axis. We call a the x-coordinate and b the y-coordinate. Points correspond to ordered pairs of numbers (x, y). A graph is an illustration of all the points whose coordinates satisfy an equation. An example of the graph y = x + 2 is given below. It intersects the x-axis at the point (-2, 0) and intersects the y-axis at the point (0, 2).

Page 2 Linear Equations Review Notes. Two points determine a line. How this is achieved: Let (x, y) and (x_0, y_0) be two distinct points. Note that the slope is m = (y - y_0) / (x - x_0). Now, we will multiply both sides by x-x_0 to get y - y_0 = m(x - x_0). We add y_0 to both sides to get y = m(x - x_0) + y_0. Now, y = mx + y_0 - mx_0. By letting b = y_0 - mx_0 we get y = mx + b. To find the x-intercept let y = 0. Example: find the x-intercept of y = 3x + 15. Note that 0 = 3x + 15 implies that -15 = 3x implies that x = -5. The x-intercept is -5. What is the x-intercept of y = 4x - 8? To find the y-intercept, let x = 0.

Page 3 Linear Equations Review Notes. Example: Find the y-intercept of y = -2x + 8. Note that putting x = 0 in y = -2x + 8 yields y = -2(0) + 8 = 8. The y-intercept is 8. What is the y-intercept of y = 5x + 7? The slope-intercept form of the equation of a line is y = mx + b where m is the slope and b is the y-intercept. Example: For y = 5x + 6, 5 is the slope and 6 is the y-intercept. If given Ax + By = C where B is not 0 then we can convert to slope-intercept form: Ax + By = C implies that By = -Ax + C implies that y = -(A/B)x + (C/B). Example: For 3x + 7y = 2, we have 3x + 7y = 2 implies that 7y = -3x + 2 implies that y = -(3/7)x + (2/7). Convert -2x + 10y = 1 to slope-intercept form. Next, we will put all of this together to graph linear equations. Equation of a horizontal line: y = k, where k is a constant. Example: y = 2. Graph with the horizontal line y = 2 shown below in the Cartesian plane going through the point (0, 2).

Page 4 Linear Equations Review Notes. Equation of a vertical line: x = c where c is a constant. Example x = 3 is shown below in the Cartesian plane where the vertical line x = 3 intersects the x-axis at the point (3, 0). Example: Graph 4x + 8y = 12. Note that 4x + 8y = 12 implies that 8y = -4x + 12 implies that y = -(1/2)x + (3/2). For the x-intercept: 0 = -(1/2)x + (3/2) implies that (1/2)x = 3/2 implies that x = 3. For the y-intercept, put x = 0 into y = -(1/2)x + (3/2) to get y = 3/2. So, (3, 0) and (0, 3/2) are points on 4x + 8y = 12. Draw a line between these points in the Cartesian plane and label the line 4x + 8y = 12. Next, graph x - 2y = 4. For the x-intercept put y = 0 into x - 2y = 4 to get x = 4. For the y-intercept, put x = 0 into x - 2y = 4 to get -2y = 4 implies that y = -2. Thus, (4,0) and (0, -2) are two points on the line x - 2y = 4. Draw the line through these points to get the graph.

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