MATH 209 - Symbolic, Graph, Numerical Solutions

by TANGIE SUMMERS
(RALEIGH, NC)

12X+2Y=24
-12X+4Y=-24

Find the symbolic, graph, numerical solutions.

Comments for MATH 209 - Symbolic, Graph, Numerical Solutions

 Apr 11, 2011 MATH 209 - Symbolic, Graph, Numerical Solutions by: Staff -------------------------------------------------------- Part II 3. Numerical Solution 12X+2Y=24 -12X+4Y=-24 Add the two equations together 12X + 2Y = 24 + (-12X + 4Y = -24) ----------------------- 12X + 2Y - 12X + 4Y = 24 - 24 12X + 2Y - 12X + 4Y = 24 - 24 12X + 2Y - 12X + 4Y = 0 Combine like terms 12X - 12X + 2Y + 4Y = 0 (12X - 12X) + (2Y + 4Y) = 0 (0) + (2Y + 4Y) = 0 0 + (2Y + 4Y) = 0 (2Y + 4Y) = 0 (6Y) = 0 6Y = 0 Divide each side of the equation by 6 6Y/6 = 0/6 6Y/6 = 0 Y*(6/6) = 0 Y*(1) = 0 Y*1 = 0 Y = 0 Substitute 0 for Y in one of the original equations, then solve for X 12X + 2Y = 24 12X + 2*0 = 24 12X + 0 = 24 Divide each side of the equation by 12 12X = 24 12X/12 = 24/12 12X/12 = 2 X*(12/12) = 2 X*(1) = 2 X*1 = 2 X = 2 The final answer is: X = 2, Y = 0 This is exactly the same answer obtained from plotting both functions and reading the coordinate pair (2,0) from the point the two graphs intersect. Check the work. Substitute 2 for x and 0 for y in the original two equations. 12X + 2Y = 24 12*2 + 2*0 = 24 12*2 + 0 = 24 12*2 = 24 24 = 24, correct -12X + 4Y = -24 -12*2 + 4*0 = -24 -12*2 + 0 = -24 -12*2 = -24 -24 = -24, correct The values of x = 2 and y = 0 are valid solutions in both of the original equations. This verifies that these values are correct solutions. Thanks for writing. Staff www.solving-math-problems.com

 Apr 11, 2011 MATH 209 - Symbolic, Graph, Numerical Solutions by: Staff The question: by TANGIE SUMMERS (RALEIGH, NC) 12X+2Y=24 -12X+4Y=-24 Find the symbolic, graph, numerical solutions. The answer: Part I 1. Symbolic Solution 1st equation: 12X + 2Y=24 12X + 2Y=24 Subtract 12X from each side of the equation 12X - 12X + 2Y=24 - 12X 0 + 2Y=24 - 12X 2Y = 24 - 12X 2Y = -12X + 24 Divide each side of the equation by 2 2Y/2 = (-12X + 24 )/2 Y*(2/2) = (-12X + 24 )/2 Y*(1) = (-12X + 24 )/2 Y*1 = (-12X + 24 )/2 Y = (-12X + 24 )/2 Y = -12X/2 + 24/2 Y = X*(-12/2) + 24/2 Y = X*(-6) + 24/2 Y = -6X + 24/2 Y = -6X + 12 2nd equation: -12X + 4Y = -24 12X + 4Y = -24 Add 12X to each side of the equation -12X + 12X + 4Y = -24 + 12X 0 + 4Y = -24 + 12X 4Y = -24 + 12X 4Y = 12X - 24 Divide each side of the equation by 4 4Y/4 = (12X - 24 )/4 Y*(4/4) = (12X - 24 )/4 Y*(1) = (12X - 24 )/4 Y*1 = (12X - 24 )/4 Y = (12X - 24 )/4 Y = 12X/4 - 24/4 Y = X*(12/4) - 24/4 Y = X*(3) - 24/4 Y = X*3 - 24/4 Y = 3X - 24/4 Y = 3X - 6 The final symbolic solutions are: 1st equation: 12X + 2Y=24 Symbolic solution to 1st equation: Y = -6X + 12 2nd equation: -12X + 4Y = -24 Symbolic solution to 2nd equation: Y = 3X - 6 2. Graphical Solution The easiest way to graph these equations is to prepare a table of (x,y) data points, and then plot the points on an x-y axis. (You can also use the slope intercept form of each equation and plot 1 (x,y) data point and the slope.) Compute four data points for the 1st equation: Y = -6X + 12 If X = 0, then Y = (-6)*(0) + 12 = 12 If X = 1, then Y = (-6)*(1) + 12 = 6 If X = 2, then Y = (-6)*(2) + 12 = 0 If X = 3, then Y = (-6)*(3) + 12 = -6 The four data points for the 1st equation are: (x,y) (0,12) (1,6) (2,0) (3,-6) When these points are plotted, they form a straight line (click link to view): http://www.solving-math-problems.com/images/math209-graph-equation-1.png Compute four data points for the 2nd equation: Y = 3X - 6 If X = 0, then Y = 3*(0) - 6 = -6 If X = 1, then Y = 3*(1) - 6 = -3 If X = 2, then Y = 3*(2) - 6 = 0 If X = 3, then Y = 3*(3) - 6 = 3 The four data points for the 2nd equation are: (x,y) (0,-6) (1,-3) (2,0) (3,3) When these points are plotted, they also form a straight line (click link to view): http://www.solving-math-problems.com/images/math209-graph-equation-2.png When both equations are plotted together, they intersect. At the point of intersection, the coordinate pairs for both equations (Y = -6X + 12 and Y = 3X – 6) are equal. These coordinates are: x = 2, y = 0. The coordinate pair at the point of intersection (2,0) is the solution to linear system (click link to view): http://www.solving-math-problems.com/images/math209-graph-equations.png The graph shows that the solution to the linear system (both equations) is: X = 2 Y = 0 --------------------------------------------------------