Problem 50 - Prealgebra

Question

Solve each equation. Check your solution $$15 s=75$$

Step-by-Step Solution

Answer

The solution is \(s = 5\).

Step 1: Understand the Equation

The equation given is in the form of a linear equation: \(15s = 75\). This equation states that 15 times the variable \(s\) equals 75.

Step 2: Solve for the Variable

To find the value of \(s\), we need to isolate \(s\) on one side of the equation. This can be done by dividing both sides of the equation by 15.

Step 3: Divide Both Sides

Divide both sides of the equation by 15 to solve for \(s\): \[s = \frac{75}{15}\] Calculate the division on the right side.

Step 4: Simplification

Calculate \(\frac{75}{15}\) = 5. So, \(s = 5\).

Step 5: Check the Solution

To verify the solution, substitute \(s = 5\) back into the original equation: \[15s = 75 \] Replace \(s\) with 5: \[15 \times 5 = 75\] Calculate the left side to confirm that it equals 75.

Step 6: Confirmation

Since \(15 \times 5 = 75\) holds true, the solution \(s = 5\) is correct.

Key Concepts

Step-by-Step Solution

A step-by-step solution helps break down a problem into manageable parts. This method is particularly useful when solving linear equations, like our original equation, \( 15s = 75 \).
Breaking the problem down into smaller steps can make complex equations much easier to handle and understand. In our example:

• Identify the goal: Find the value of \( s \) so that the equation is true.
• Isolate the variable: We need to have \( s \) by itself on one side of the equation to find its value.
• Simplify the equation: Divide both sides by the number that multiplies \( s \) (in this case, 15).
• Calculate the result: Perform the arithmetic to find the value of \( s \).

This structured method ensures that every detail is accounted for, reducing the risk of error and increasing understanding.

Checking Solutions

Once you solve an equation, it's essential to check your solution. This step ensures that what you've calculated is actually correct. To check a solution:

• Substitute back: Replace the variable in the original equation with your calculated solution.
• Calculate both sides: Perform the arithmetic to ensure both sides of the equation equal each other.
• Confirm equality: If both sides are indeed equal, your solution is verified!

For example, after finding \( s = 5 \), substitute it back into \( 15s = 75 \), resulting in \( 15 \times 5 = 75 \). This confirms our solution is correct, because both sides match.
Checking ensures accuracy and boosts confidence in your problem-solving skills.

Algebraic Equations

Algebraic equations are mathematical statements that show the equality between two expressions. They often include variables that represent unknown values, which we need to find. In algebra, you might encounter different forms of equations:

• Linear equations: These are the simplest form and look like \( ax + b = c \).
• Quadratic equations: These include terms like \( ax^2 + bx + c = 0 \).
• Polynomial equations: These can have variables raised to higher powers.

The basic principle of solving an algebraic equation is to perform the same operation on both sides, maintaining the balance or equality.
For linear equations, this often involves using addition, subtraction, multiplication, or division to isolate the variable. Understanding these concepts will help you tackle a wide range of problems effectively.

Understanding Equations

Understanding equations means knowing the core principles that define them. At its core, an equation is about balance. Each side of an equation must equal the other in value. When you have the equation \( 15s = 75 \), it's saying that 15 times some value (\( s \)) equals 75.To solve the equation:

• You look for ways to isolate \( s \).
• This involves reversing operations (like using division when dealing with multiplication).
• The aim is to get \( s \) alone on one side.

Each operation and step must maintain the stability of the equation.
Recognizing that operations are reversible is crucial for solving equations. This understanding unlocks the ability to manipulate and solve problems systematically.