Problem 23 - Algebra 1

Question

Is the number given a solution of the equation? $$12-2 y=6 ; 4$$

Step-by-Step Solution

Answer

No, 4 is not a solution to the equation \(12 - 2y = 6\).

Step 1: Understand the equation

The provided equation is \(12 - 2y = 6\). We have to check if \(y = 4\) is a solution to this equation.

Step 2: Substitute the given value

Substitute \(y = 4\) into the equation. So, the equation becomes \(12 - 2 * 4 = 6\). After multiplication, the equation becomes \(12 - 8 = 6\).

Step 3: Simplify the equation

Resolve the subtraction operation on left hand side and the equation becomes \(4 = 6\).

Step 4: Verify

The simplified equation gives \(4 = 6\). Since 4 is not equal to 6, it means that \(y = 4\) is not a solution to the equation \(12 - 2y = 6\).

Key Concepts

Solution of Equations

Solving equations is a fundamental concept in algebra, where the goal is to find the value of the variable that makes the equation true. In the given exercise, we are asked to verify whether the number 4 is a solution to the equation \(12 - 2y = 6\).
To check if a number is a solution, substitute it into the equation in place of the variable. If the equation holds true after the substitution, the number is indeed a solution.
However, in this example, substituting \(y = 4\) did not satisfy the equation, meaning it is not a solution. Understanding how to verify potential solutions helps in mastering equation-solving skills.

Substitution Method

The substitution method is a technique used for solving equations by replacing variables with given values. It's straightforward and can be applied to determine if a specific number is a solution for an equation.
In the provided problem, we used the substitution method by inserting \(y = 4\) into the equation \(12 - 2y = 6\), transforming it into \(12 - 2 \times 4 = 6\).

• This approach is useful when you have a candidate value for a variable and want to determine if it satisfies the entire equation.
• If, after substituting, both sides of the equation are equal, the number is a solution.
• On the other hand, if they aren't, like in this case (\(4 eq 6\)), it indicates that the substitution for \(y\) was incorrect.

Thus, substitution is a key method for verifying solutions effectively and is foundational for solving more complex systems of equations.

Algebraic Expressions

Algebraic expressions form the building blocks of equations. They consist of numbers, variables, and operations (like addition, subtraction, etc.).
In our equation \(12 - 2y = 6\), the expression \(12 - 2y\) is algebraic because it combines constants, the variable \(y\), and subtraction.

• Understanding how to manipulate these expressions through operations like addition, subtraction, multiplication, and division is crucial.
• When solving equations, algebraic expressions are used to balance both sides to find the variables' values.
• In the case of our example, simplifying \(12 - 2 \times 4 = 6\) was an essential step to determining if \(y = 4\) was valid.

Developing a strong grasp of how algebraic expressions work paves the way for success in various algebraic problems and understanding the solution process thoroughly.