Questions: equation has infinitely (4x+3)/3 = 4x+5

Transcript text: equation has infinitely \[ \frac{4 x+3}{3}=4 x+5 \]

Solution

Solution Steps

To determine if the given equation has infinitely many solutions, we need to simplify and solve the equation. If the equation simplifies to a true statement (like \(0 = 0\)), it has infinitely many solutions. Otherwise, it has a unique solution or no solution.

Step 1: Simplify the Equation

Given the equation: \[ \frac{4x + 3}{3} = 4x + 5 \] we first simplify it by multiplying both sides by 3 to eliminate the fraction: \[ 4x + 3 = 3(4x + 5) \]

Step 2: Expand and Combine Like Terms

Next, we expand the right-hand side: \[ 4x + 3 = 12x + 15 \] Then, we move all terms involving \(x\) to one side and constant terms to the other side: \[ 4x - 12x = 15 - 3 \] \[ -8x = 12 \]

Step 3: Solve for \(x\)

We solve for \(x\) by dividing both sides by -8: \[ x = \frac{12}{-8} = -\frac{3}{2} \]