Final answer:
The absolute value equation |2x-4|=|-5x+1| can be rewritten as two linear equations, 2x-4=-5x+1 and 2x-4=5x-1, which can be solved to find the values of x. There are two possible solutions: x=5/7 or x=-1.
Step-by-step explanation:
To rewrite the absolute value equation |2x-4|=|-5x+1| as two systems of linear equations, we need to consider the definition of absolute value. The absolute value of a number is the distance of that number from zero on the number line, without considering which direction from zero it is. Therefore, if the absolute values of two expressions are equal, it means the two expressions are either the same or they are negatives of each other.
We can express this as two separate linear equations:
2x - 4 = -5x + 1 (where both expressions have the same value)
2x - 4 = 5x - 1 (where one expression is the negative of the other)
To solve these linear equations, you simplify each one to find the value of x. For the first equation:
- Add 5x to both sides to get 7x - 4 = 1.
- Add 4 to both sides to get 7x = 5.
- Divide both sides by 7 to find x = 5/7.
For the second equation:
- Subtract 2x from both sides to get -4 = 3x - 1.
- Add 1 to both sides to get -3 = 3x.
- Divide both sides by 3 to find x = -1.
So, the possible solutions for the equation are x = 5/7 and x = -1.