Problem 46 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether the given number is a solution of the equation. $$50-y=20 ; 30$$

Step-by-Step Solution

Answer

Yes, 30 is a solution to the given equation.

Step 1: Identify the Equation and the Number to Test

The equation provided is \(50-y=20\) and the number provided to test is 30. This essentially means that we want to evaluate if the value of \(y=30\) is a solution to the equation.

Step 2: Substitute the Number into the Equation

To determine if 30 is a solution to the equation, substitute 30 for \(y\) in the equation, which gives us \(50-30=20\).

Step 3: Simplify the Left Side and Check Equality

Simplify the left side of the equation to: \(50-30=20\), which simplifies further to: \(20=20\). Since both sides of the equation are equal, 30 is a solution to the equation.

Key Concepts

Equation Solving

Equation solving involves finding the value of a variable that makes an equation true. An equation is a mathematical statement that asserts the equality of two expressions. For example, in the equation \(50-y=20\), we aim to find the value of \(y\) that makes both sides equal. This process can seem complex, but it's often broken down into manageable steps:

• First, identify which equation you are dealing with.
• Next, determine what variable or number needs to be found.
• Finally, use mathematical operations to isolate the variable and solve the equation.

Understanding equations is fundamental in algebra. It requires recognizing the elements involved—constants, variables, and operations such as addition or subtraction.

Substitution Method

The substitution method is a common technique in algebra for testing if a given number is a solution to an equation. This involves replacing the variable with the number in question and checking if the resulting equation holds true.
In our specific example, the substitution method is used to test whether \(y=30\) satisfies the equation \(50-y=20\). By substituting 30 for \(y\), we create a new equation: \(50-30=20\). This method lets us focus on calculating the expression without guessing if a number works.
The substitution method is valuable because:

• It provides a direct way to verify solutions.
• It helps minimize errors in more complex equations.

Knowing how to correctly perform substitution can streamline solving equations significantly.

Solution Verification

Verifying solutions is about confirming that our substitutions and calculations were correct. In algebra, after performing operations like substitution, the next step is to simplify the equation to see if a true statement is produced.
For instance, after substituting \(y = 30\) into the equation \(50-y=20\), the next step is to ensure it simplifies to \(20=20\). The left side of the equation becomes \(50-30\), which simplifies to 20.
Verification is crucial because it:

• Ensures that the solution meets the original equation's conditions.
• Helps identify calculation errors early on.

Being meticulous with verification means you can confidently conclude whether a number is a valid solution.