Problem 140 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ -3 z=\frac{2 z}{5} $$

Step-by-Step Solution

Answer

z = 0

Step 1: Clear Fraction by Multiplying

The given equation is \[-3z = \frac{2z}{5}\]To eliminate the fraction, multiply each term by 5, giving:\[5(-3z) = 5\left(\frac{2z}{5}\right)\]This simplifies to:\[-15z = 2z\]

Step 2: Combine Like Terms

We need to collect all the terms involving \(z\) on one side of the equation. Subtract \(2z\) from both sides:\[-15z - 2z = 0\]Simplifying, we get:\[-17z = 0\]

Step 3: Solve for z

To isolate \(z\), divide both sides of the equation by -17:\[\frac{-17z}{-17} = \frac{0}{-17}\]Thus, we find:\[z = 0\]

Key Concepts

Clearing Fractions

Clearing fractions from an equation is a key step to simplify and solve it. When you have a fraction, it can complicate your calculations, so you want to get rid of it. In our example, the equation is \(-3z = \frac{2z}{5}\). To clear the fraction, you multiply each term in the equation by the denominator of the fraction, which is 5 in this case.

This step involves:

• Identifying the denominator in the fraction—here it is 5.
• Multiplying every term in the equation by this denominator.

So, the equation \(-3z = \frac{2z}{5}\) becomes \(5(-3z) = 5\left(\frac{2z}{5}\right)\). The fraction disappears, leaving you with \(-15z = 2z\). Clearing fractions like this can help you focus on solving the core equation without the added complexity of fractions.

Combining Like Terms

Once the fractions are cleared, the next step is to simplify the equation by combining like terms. Like terms are terms that contain the same variables raised to the same power. In the equation \(-15z = 2z\), both terms are like terms because they both involve the variable \(z\).

To combine them, follow these steps:

• Move all terms involving the variable you are solving for (in this case, \(z\)) to one side of the equation.
• Subtract \(2z\) from both sides: \(-15z - 2z = 0\).

This simplifies to \(-17z = 0\). By combining like terms, you consolidate the equation, making it easier to find the solution.

Isolating Variables

The final step in solving a linear equation is to isolate the variable. In our scenario, after combining like terms, we have \(-17z = 0\). The goal is to have \(z\) by itself on one side of the equation. By isolating the variable, you directly solve for its value.

Here's how you isolate the variable \(z\):

• Divide each side of the equation by the coefficient of the variable. Here, the coefficient is \(-17\).
• Perform the division: \(\frac{-17z}{-17} = \frac{0}{-17}\).

This gives \(z = 0\). Isolating variables is a crucial step in solving equations, as it leads directly to the solution.