Certainly! Let's go through each of the given equations step-by-step to determine the solution type: one solution, infinitely many solutions, or no solution.
### 1. Equation: \( 5(x-2) = 5x - 7 \)
Let's expand and simplify both sides:
[tex]\[ 5(x - 2) = 5x - 10 \][/tex]
The equation now is:
[tex]\[ 5x - 10 = 5x - 7 \][/tex]
Subtract \( 5x \) from both sides:
[tex]\[ -10 = -7 \][/tex]
This is a contradiction. Therefore, this equation has No Solution.
### 2. Equation: \( -3(x-4) = -3x + 12 \)
Let's expand and simplify both sides:
[tex]\[ -3(x - 4) = -3x + 12 \][/tex]
[tex]\[ -3x + 12 = -3x + 12 \][/tex]
Both sides are identical. Therefore, this equation has Infinitely Many Solutions.
### 3. Equation: \( 4(x+1) = 3x + 4 \)
Let's expand and simplify both sides:
[tex]\[ 4(x + 1) = 4x + 4 \][/tex]
Now, the equation is:
[tex]\[ 4x + 4 = 3x + 4 \][/tex]
Subtract \( 3x \) from both sides:
[tex]\[ x + 4 = 4 \][/tex]
Subtract 4 from both sides:
[tex]\[ x = 0 \][/tex]
So, this equation has One Solution.
### 4. Equation: \( -2(x-3) = 2x - 6 \)
Let's expand and simplify both sides:
[tex]\[ -2(x - 3) = -2x + 6 \][/tex]
The equation now is:
[tex]\[ -2x + 6 = 2x - 6 \][/tex]
Add \( 2x \) to both sides:
[tex]\[ 6 = 4x - 6 \][/tex]
Add 6 to both sides:
[tex]\[ 12 = 4x \][/tex]
Divide by 4:
[tex]\[ x = 3 \][/tex]
So, this equation has One Solution.
### 5. Equation: \( 6(x+5) = 6x + 11 \)
Let's expand and simplify both sides:
[tex]\[ 6(x + 5) = 6x + 30 \][/tex]
The equation now is:
[tex]\[ 6x + 30 = 6x + 11 \][/tex]
Subtract \( 6x \) from both sides:
[tex]\[ 30 = 11 \][/tex]
This is a contradiction. Therefore, this equation has No Solution.
### Summary
Let's sort the equations based on their solutions:
- Infinitely Many Solutions:
- \( -3(x-4) = -3x + 12 \)
- No Solution:
- \( 5(x-2) = 5x - 7 \)
- \( 6(x+5) = 6x + 11 \)
- One Solution:
- \( 4(x+1) = 3x + 4 \)
- [tex]\( -2(x-3) = 2x - 6 \)[/tex]
