Solve Using a Matrix by Elimination 1/(x-5)-3/(x+6)=0 , 3/(x+7)-1/(y-3)=0

Problem

{[1/(x−5)−3/(y+6)=0],[3/(x+7)−1/(y−3)=0])

Solution

1. Rearrange the equations to isolate the terms on opposite sides of the equals sign.

1/(x−5)=3/(y+6)

3/(x+7)=1/(y−3)

2. Cross-multiply each equation to eliminate the fractions and create a system of linear equations.

y+6=3*(x−5)

3*(y−3)=x+7

3. Distribute the constants and move all variables to the left side and constants to the right side.

y+6=3*x−15⇒−3*x+y=−21

3*y−9=x+7⇒−x+3*y=16

4. Write the system as an augmented matrix.

[[−3,1,|,−21],[−1,3,|,16]]

5. Perform row operations to reach row-echelon form. Multiply the second row by −3

[[−3,1,|,−21],[3,−9,|,−48]]

6. Add the first row to the second row to eliminate the x term in the second row.

[[−3,1,|,−21],[0,−8,|,−69]]

7. Solve for y using the second row.

−8*y=−69

y=69/8

8. Substitute the value of y back into the first equation to solve for x

−3*x+69/8=−21

−3*x=−21−69/8

−3*x=−168/8−69/8

−3*x=−237/8

x=79/8

Final Answer

(x,y)=(79/8,69/8)

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