Problem 64 - Algebra 1: Concepts and Skills

Question

Solve the equation if possible. Check your solution. $$ 9-5 z=-8 z $$

Step-by-Step Solution

Answer

The original solution is incorrect and leads to a contradiction when substituted back into the original equation. There is no solution for the given exercise, satisfying the initial equation.

Step 1: Rearrange the equation

Begin by rearranging the equation to group the \(z\) terms together. Do this by adding \(5z\) to each side of the equation. This gets us \(9 = 3z\).

Step 2: Isolate the variable \(z\)

Next, divide each side of the equation by 3, to isolate \(z\). This gives the solution \(z=3\).

Step 3: Check the solution

Finally, confirm the solution by substituting \(z=3\) into the original equation and checking if both sides of the equation are equal. Substituting \(z=3\) into the original equation \(9-5z=-8z\) we get \(9-5*3=-8*3 \); which simplifies to \(9-15=-24\) and further simplifies to \(-6=-24\). This is not a correct equation, therefore, our original solution is wrong. There is a mistake in our work. The mistake is in the first step when adding \(5z\) to both sides, the correct equation should be \(9=13z\) not \(9=3z\). In this case there is no solution since adding the same number to both sides of the equation does not lead to an equivalent equation.

Key Concepts

Variable Isolation

Variable isolation is an essential step in solving equations. It means manipulating the equation so that one side is left with just the variable in question and the other side is left with its value. This step often involves addition, subtraction, multiplication, or division operations. For example, if you have an equation like \(9 = 13z\), you can isolate \(z\) by dividing both sides by 13 to find that \(z = \frac{9}{13}\).
This operation must be done carefully to ensure that any operations performed on one side of the equation are mirrored on the other side. This maintains the balance of the equation, a crucial factor in effective equation solving. Always remember that after isolating the variable, it's advisable to check the solution back in the original equation to ensure accuracy.

Checking Solutions

Once a solution is derived, the next step is to verify its correctness by substituting the value back into the original equation. Checking your solution is a vital part of solving equations, as a mistake in earlier steps can lead to incorrect answers. Consider the equation \(9 - 5z = -8z\).
If we've isolated \(z\) and found it, we substitute back into the original equation to confirm its validity. Finding a contradiction—such as substituting and ending with \(-6 = -24\) after working through the operations—reveals an error was made. This step can avoid errors and helps understanding of the solution process, ensuring the steps taken were valid.

Rearranging Equations

Rearranging equations is often the initial step when dealing with variable isolation. The aim is to arrange the terms so that variables are on one side of the equation and constants are on the other. In the equation \(9 - 5z = -8z\), rearranging begins by adding \(5z\) to both sides, aiming to group the \(z\) terms together.
This results in the equation \(9 = -8z + 5z\), simplifying to \(9 = -3z\). Failure to perform proper rearranging, as shown in the original step attempting \(9 = 3z\), leads to mistakes. Rearranging correctly prepares the equation for variable isolation and ensures the resulting steps will yield the correct solution.