How many solutions does this equation have?
$9 z+10=1-3(5-3 z)$
We are given the equation \(9z + 10 = 1 - 3(5 - 3z)\). Our goal is to determine the number of solutions for \(z\). A linear equation in the form \(Az + B = Cz + D\) generally has three possibilities: one unique solution if \(A eq C\), infinitely many solutions if \(A = C\) and \(B = D\), or no solution if \(A = C\) and \(B eq D\). We will simplify the given equation to identify which case applies here.
First, we expand the expression on the right side of the equation:
\[1 - 3(5 - 3z) = 1 - (3 \times 5) + (3 \times 3z) = 1 - 15 + 9z = 9z - 14.\]
Now, we substitute this back into the original equation:
\[9z + 10 = 9z - 14.\]
Next, we look at the terms involving the variable \(z\) and the constant terms on both sides of the equation. We have:
\[9z + 10 = 9z - 14.\]
If we subtract \(9z\) from both sides, we get:
\[10 = -14.\]
Since \(10 = -14\) is a mathematically impossible statement (a contradiction), the equation has no value of \(z\) that makes it true. Therefore, there are no solutions.
Imagine you are trying to find two numbers that balance a scale. If the weight on one side is always exactly 24 units lighter than the other side, no matter what weight you add to both, the scale will never balance. This is exactly what happened with our equation: because the variable parts cancel out perfectly, the remaining constants show a fundamental inequality that can never be reconciled, resulting in zero solutions. This concept is essential in physics and engineering for identifying impossible design constraints.