Question

Determine whether each equation has one solution, no solution, or infinite solutions.

Equation
1. 4(x+5)+4=1/2(8x+48)
2. 5(x-2)-x=5x+2
3. 2(4-x)=-2(1+x)-2
4.3(8-4)-x=6(4x-1)-6

Answer 1

Equation 1 has infinitely many solutions, equation 2 has one solution, equation 3 has no solution, and equation 4 has one solution.

Step-by-step explanation:

To determine whether each equation has one solution, no solution, or infinite solutions, we will simplify and solve each equation step by step:

1. 4(x+5)+4=1/2(8x+48)
   Simplifying both sides of the equation gives us 4x + 20 + 4 = 4x + 24. Combining like terms gives us 4x + 24 = 4x + 24. Subtracting 4x from both sides eliminates the variable, leaving us with 24 = 24. Since 24 is equal to 24, this equation has infinitely many solutions.
2. 5(x-2)-x=5x+23
   Simplifying both sides of the equation gives us 5x - 10 - x = 5x + 23. Combining like terms gives us 4x - 10 = 5x + 23. Subtracting 4x from both sides gives us -10 = x + 23. Subtracting 23 from both sides gives us -33 = x. Therefore, this equation has one solution, x = -33.
3. 2(4-x)=-2(1+x)-2
   Simplifying both sides of the equation gives us 8 - 2x = -2 - 2x - 2. Combining like terms gives us 8 - 2x = -4 - 2x. Adding 2x to both sides eliminates the variable, leaving us with 8 = -4. Since 8 is not equal to -4, this equation has no solution.
4. 3(8-4)-x=6(4x-1)-6
   Simplifying both sides of the equation gives us 12 - x = 24x - 6 - 6. Combining like terms gives us 12 - x = 24x - 12. Adding x to both sides eliminates the variable, leaving us with 12 = 25x - 12. Adding 12 to both sides gives us 24 = 25x. Dividing both sides by 25 gives us x = 24/25. Therefore, this equation has one solution, x = 24/25.