Problem 14 - Algebra 1

Question

CHECKING SOLUTIONS OF EQUATIONS Check whether the given number is a solution of the equation. $$3 b+1=13 ; 4$$

Step-by-Step Solution

Answer

Yes, 4 is a solution to the equation.

Step 1: Substitution

Replace \(b\) in the equation with the given number. This results in \(3(4) + 1\).

Step 2: Simplification

Simplify the expression to see if it evaluates to 13. \(3 * 4 = 12\), and \(12 + 1 = 13\). Thus, when simplified, the equation is \(13 = 13\).

Step 3: Verification

Determine if the simplified equation is true. Since \(13 = 13\) is a true statement, the input number, 4, is a solution to the equation.

Key Concepts

Substitution in Equations

Substitution is a useful tool in the world of mathematics. It involves replacing a variable in an equation with a given number to determine if it satisfies the equation.
You start by identifying the variable in your equation. In our example, the equation is \(3b + 1 = 13\). Here, the variable is \(b\).
- Next, substitute the variable with the number provided in the problem. In our case, we replace \(b\) with \(4\), resulting in \(3(4) + 1\).By doing this, you're essentially checking to see if using this number makes the equation true. If it does, you know this number is a solution.

Simplification of Expressions

Simplification is a crucial step after substitution to verify the viability of an equation. Once you've substituted a number for the variable, your next goal is to simplify the mathematical expression as much as possible.Here, after substituting \(4\) for \(b\), the equation becomes \(3(4) + 1\). First, multiply \(3\) by \(4\), resulting in \(12\).
Adding \(1\) to \(12\) then gives us \(13\).- The process of simplification breaks down complex expressions into more understandable terms.- It's important to follow the order of operations (parentheses, exponents, multiplication and division, addition and subtraction) to simplify correctly. When simplified correctly, the equation should now clearly show a statement like \(13 = 13\), making it easy to compare both sides of the equation.

Verification of Solutions

Verification is the final check to confirm whether a substituted value truly satisfies an equation. This step checks the accuracy and validity of the solution:- After the simplification process, you evaluate the left-hand side of the equation against the right-hand side.- For the equation \(3b + 1 = 13\) with \(b = 4\), if both sides equal \(13\), then the given number is indeed a solution.In this case, since \(13 = 13\), this confirms that \(4\) is a solution to the equation. Verification is vital because it ensures that no errors have been made in the substitution or simplification stages. It's your final stamp of "true" or "false" for the given number as a solution.