- Combine like terms on the left side: \(17z + 9 = 4z + 5\).
- Isolate the variable by subtracting \(4z\) from both sides: \(13z + 9 = 5\).
- Solve for \(z\) by subtracting \(9\) and dividing by \(13\): \(z = -4/13\).
- The equation has exactly one solution: \(\boxed{\text{(B)}}\).
Explanation
- Simplifying the Equation
To determine the number of solutions to the equation \(3z + 9 + 14z = 4z + 5\), we first simplify the expression on the left-hand side. By combining the like terms \(3z\) and \(14z\), we obtain \(17z + 9 = 4z + 5\). This is a linear equation in one variable, \(z\). Our goal is to isolate the variable on one side and the constant on the other to determine its solutions.
- Solving for z and Concluding
Next, we subtract \(4z\) from both sides of the equation:
\[17z - 4z + 9 = 5\]
\[13z + 9 = 5\]
Then, we subtract \(9\) from both sides:
\[13z = 5 - 9\]
\[13z = -4\]
Finally, dividing by \(13\), we find the solution:
\[z = -\frac{4}{13}\]
Since we have found a single, specific value for \(z\), the equation has exactly one solution.
Examples
Linear equations are frequently used in everyday scenarios where you need to find an unknown value. For example, if you are comparing two different cell phone plans, one might have a fixed monthly fee plus a variable cost per gigabyte, and another might have a different setup. Setting these two expressions equal allows you to find the exact usage amount where both plans cost the same, helping you make a more informed choice based on your typical data consumption.