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MATH 209 - Symbolic, Graph, Numerical Solutions

by TANGIE SUMMERS
(RALEIGH, NC)











































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

Find the symbolic, graph, numerical solutions.

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Apr 11, 2011
MATH 209 - Symbolic, Graph, Numerical Solutions
by: Staff


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Part II

3. Numerical Solution

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

Add the two equations together

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

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