Problem 45 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether the given number is a solution of the equation. $$30-y=10 ; 20$$

Step-by-Step Solution

Answer

Yes, the number 20 is a solution to the equation \(30 - y = 10\).

Step 1: Re-arrange the equation

Start by re-arranging the equation to solve for \(y\). To do this, one would subtract 30 from both sides of the equation, resulting in the equation: \(-y = 10 - 30\)

Step 2: Simplify the equation

Next, subtract 10 from 30 to get -20, so the simplified equation would be \(-y = -20\)

Step 3: Multiply Through by -1

Multiply the whole equation by -1 so as to get \(y\) alone on one side. We find that \(y = 20\)

Step 4: Compare with the given number

Compare this \(y\) value with the given number. Result is \(20 = 20\). As the result indicates, the given number 20 is indeed a solution to the equation.

Key Concepts

Solving Equations

When it comes to solving equations, we are essentially finding values that make the equation true. In this scenario, our task is to verify if a number is a given solution to an algebraic equation. Solving equations involves several steps to determine if the candidate value satisfies the equation. The equation and the potential solution were given as: \(30 - y = 10; 20\). Our job is to check whether substituting \(y = 20\) into the equation leads to a true statement. If it does, 20 is indeed a solution.

In the original step-by-step solution:

• Start from the full equation \(30 - y = 10\).
• Then manipulate the equation so it can show if your assumption of \(y = 20\) is correct or not.

This whole process is to ensure the logical correctness of finding the solution. The whole essence of solving equations lies in methodically transforming and analyzing the equation until you derive the correct answer.

Variable Isolation

Variable isolation is a key step in solving equations because it helps you get the unknown, or variable, by itself on one side of the equation. For the equation \(30 - y = 10\), the variable we want to isolate is \(y\). To isolate \(y\), we need to perform operations that will leave \(y\) on one side without other numbers or operations attached.

Here's how it works:

• First, subtract 30 from both sides: \(-y = 10 - 30\). The equation becomes \(-y = -20\). This eliminates the constant term on the side with \(y\).
• Next, since \(y\) is negative, multiply through by -1: \(y = 20\).

By following these steps, \(y\) is isolated on one side, providing a clearer path to check if the solution is valid or correct. Isolating variables is a fundamental skill in algebra that sets the stage for solving more complicated equations.

Equation Simplification

Equation simplification goes hand-in-hand with the isolating of variables. It involves reducing the complexity of the equation by performing operations that simplify each side. This helps make the solution process smoother and faster.

The original problem began with simplifying the equation: \(30 - y = 10\). The goal is to reduce this expression to its simplest form.

• Subtracting 30 from both sides gives \(-y = 10 - 30\), leading to \(-y = -20\).
• The subtraction of complex parts ensures that the variables and constants are in the simplest possible form.
• Multiplying through by -1 results in a cleaner equation: \(y = 20\).

Simplifying equations is about finding the easiest path to solve for the variable. This concept is not only useful in algebraic equations but also essential for simplifying any mathematical problem.