To create an equation with infinitely many solutions, the left-hand side must be equal to the right-hand side for all values of \(x\). This can be done by ensuring that both sides simplify to the same expression.
Start with the left-hand side:
\[5(3x - 4) = 15x - 20\]
Now, let's write the right-hand side as:
\[4 + Ax + B\]
We want the equation to be true for all \(x\), meaning the coefficients of \(x\) and the constant terms must match on both sides.
From the left side, we have:
- Coefficient of \(x\) is \(15\).
- Constant term is \(-20\).
For the right side \(4 + Ax + B\), we have:
- Coefficient of \(x\) is \(A\).
- The constant term is \(4 + B\).
To create an equation with infinitely many solutions, set the coefficients equal:
- \(A = 15\)
- \(4 + B = -20\)
From the second equation, you can solve for \(B\):
\[B = -20 - 4\] \[B = -24\]
Thus, we can choose \(A = 15\) and \(B = -24\).
Putting it all together, we have:
\[5(3x - 4) = 4 + 15x - 24\]
So the completed equation is:
\[5(3x - 4) = 4 + 15x - 24\]
This creates an equation that reduces to the same expression on both sides, thereby ensuring it has infinitely many solutions.