Problem 32 - Algebra 1: Concepts and Skills

Question

CHECKING SOLUTIONS OF EQUATIONS. Check to see if the given value of the variable is or is not a solution of the equation. \(9+2 t=15 ; t=12\)

Step-by-Step Solution

Answer

The value \(t=12\) is not a solution to the equation \(9 + 2t = 15\), because when replacing, the left-hand side (\(33\)) does not equal the right-hand side (\(15\)).

Step 1 - Replace the Variable with the Given Value

Replace \(t\) with \(12\) in the equation:\(9 + 2(12) = 15\)

Step 2 - Perform the Operation

Perform the multiplication operation:\(9 + 24 = 15\).

Step 3 - Simplify the Equation

Simplify the left-hand side of the equation:\(33 = 15\).

Key Concepts

Equation Solving

Equation solving is one of the fundamental skills in algebra. It involves finding the value of the variable that makes the equation true. An equation is a mathematical statement that asserts the equality of two expressions. These expressions may include numbers, variables, and arithmetic operations like addition, subtraction, multiplication, or division.
To solve equations, follow these basic steps:

• Identify the variable whose value you are seeking.
• Perform operations to isolate the variable on one side of the equation.
• Ensure that any operations you perform on one side, you must also do on the other to maintain equality.

In the given exercise, the equation is solved by substituting the variable with a given value to check if it satisfies the equation. Such exercises help to ascertain the correctness of the derivation and ensure accuracy in algebraic calculations.

Substitution in Equations

Substitution in equations involves replacing variables with specific values to evaluate the equation. This process is crucial when checking if a particular value is a solution to the given equation.
For example, in the original exercise:

• The equation provided is: \( 9 + 2t = 15 \).
• The value given for \( t \) is \( 12 \).

To check if \( t = 12 \) is a solution, replace \( t \) with \( 12 \) in the equation, which results in \( 9 + 2(12) = 15 \). Then perform the arithmetic operations to see if both sides of the equation hold true.
Substitution is a powerful technique used not only for verifying solutions but also for tackling more complex algebraic problems, where equations or systems of equations involve one or more variables.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that together represent a value or set of values. Understanding how to manipulate these expressions is central to mastering algebra.
Here are some components of algebraic expressions:

• Constants: fixed values like \( 9 \) in the expression \( 9 + 2t \).
• Variables: symbols representing unknown values, such as \( t \).
• Coefficients: numbers multiplying the variables, like \( 2 \) in \( 2t \).
• Operators: symbols denoting operations (addition, subtraction, etc.), such as \(+\) in the expression \( 9 + 2t \).

By understanding the structure of algebraic expressions, you can effectively perform operations to simplify expressions, solve equations, and verify solutions. In the context of the exercise, performing operations on the left-hand side simplifies the expression and reveals whether the two sides of the equation are equal at the given variable value.