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Extra info for An easier solution of a Diophantine problem about triangles, in which those lines from the vertices which bisect the opposite sides may be expressed rationally

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Example 6: Draw the graph of 2x + 3y = 12 by finding two random points. To do this, select a value for one variable; then substitute this into the equation and solve for the other variable. Do this a second time with new values to get a second point. Let x = 2; then find y. 2x + 3y = 12 2(2) + 3y = 12 4 + 3y = 12 3y = 8 y = 83 Therefore, the ordered pair (2, 83 ) belongs on the graph. Let y = 6; then find x. 2x + 3y = 12 2x + 3(6) = 12 2x + 18 = 12 2x = –6 x = –3 Therefore, the ordered pair (–3,6) belongs on the graph.

The y-intercept of a graph is the point at which the graph will intersect the y-axis. It will always have an x-coordinate of zero. A vertical line that is not the y-axis will have no y-intercept. One way to graph a linear equation is to find solutions by giving a value to one variable and solving the resulting equation for the other variable. A minimum of two points is necessary to graph a linear equation. Example 6: Draw the graph of 2x + 3y = 12 by finding two random points. To do this, select a value for one variable; then substitute this into the equation and solve for the other variable.

X + 3y - 4z =- 7 3x + y + 2z = 5 multiply (- 3) multiply (1) - 3x - 9y + 12z = 21 3x + y + 2z = 5 - 8y + 14z = 26 (5) Solve the system created by equations (4) and (5). - 14y + 19z =- 29 - 8y + 14z = 26 multiply (- 8) multiply (14) 112y - 152z =- 232 - 112y + 196z = 364 44z = 132 z= 3 Chapter 4: Linear Equations in Three Variables 59 Now, substitute z = 3 into equation (4) to find y. –14y + 19z = 29 –14y + 19(3) = 29 –14y + 57 = 29 –14y = –28 y=2 Use the answers from Step 4 and substitute into any equation involving the remaining variable.

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