Problem 52 - Prealgebra

Question

Solve each equation. Check your solution $$25 z=175$$

Step-by-Step Solution

Answer

The solution is \(z = 7\).

Step 1: Understanding the Equation

The equation given is a simple linear equation: \(25z = 175\). Our goal is to find the value of \(z\) that satisfies this equation.

Step 2: Isolate the Variable

To find \(z\), divide both sides of the equation by 25. This gives us: \[ z = \frac{175}{25} \]

Step 3: Simplify the Equation

Perform the division: \( \frac{175}{25} = 7 \). Therefore, \(z = 7\).

Step 4: Check the Solution

Substitute \(z = 7\) back into the original equation to verify: \(25 \times 7 = 175\). Since both sides are equal, the solution is correct.

Key Concepts

Solving Equations

When we talk about solving equations, our main goal is to find the value of the unknown variable that makes the equation true. In the equation \(25z = 175\), the variable is \(z\). To solve it, we need to perform operations that will isolate this variable, meaning getting it by itself on one side of the equation. This usually involves reversing operations that have been applied to the variable by using their opposites.

• If the variable is multiplied by a number, we divide by that number.
• If it is added to a number, we subtract that number.

In our example, \(z\) is multiplied by 25, so we divide both sides of the equation by 25 to solve it.

Checking Solutions

Once we have found a solution, like \(z = 7\), we must ensure it's correct. This is done by substituting the solution back into the original equation. For our equation \(25z = 175\), we replace \(z\) with 7 and get \(25 \times 7 = 175\).If both sides of the equation are equal, the solution is confirmed as correct. If they are not, we need to check our calculation steps again. Checking the solution helps catch any mistakes and ensure that the value we found truly satisfies the equation. It’s an essential step, especially when dealing with more complex equations.

Isolation of Variables

The process of isolating variables involves performing mathematical operations that transform an equation until the variable is alone on one side. The aim is to "undo" the operations that surround the variable.In the linear equation \(25z = 175\), \(z\) is isolated by dividing both sides of the equation by 25. This action reverses the multiplication of \(z\) by 25 and allows us to find its value.Here are a few tips when isolating variables:

• Always perform the same operation on both sides of the equation to maintain balance.
• Work step-by-step and simplify as you go.
• Reevaluate your steps if the solution seems incorrect after checking it.

This method is a key component of solving linear equations and paves the way for solving more complicated equations in the future.