Problem 63 - Algebra 1: Concepts and Skills

Question

SOLVING EQUATIONS Solve the equation. (Lesson 3.1) $$ 4+y=12 $$

Step-by-Step Solution

Answer

The solved value of y is 8.

Step 1: Identify the Equation

The given equation is \(4 + y = 12\) where it is required to solve for the variable y.

Step 2: Transposition

Transpose +4 from the left-hand side to the right-hand side, a positive term becomes negative when transposed. The equation becomes \(y = 12 - 4\).

Step 3: Evaluation

Subtract 4 from 12 to get the value of y. The resulting equation is \(y = 8\).

Key Concepts

Algebraic Expressions

Algebraic expressions are at the core of algebra and are combinations of letters and numbers using arithmetic operations like addition, subtraction, multiplication, and division. These letters, often referred to as variables, represent unknown values that can be manipulated through algebraic operations to solve problems.

For example, in the equation from our exercise, \(4 + y = 12\), \('y'\) is a variable within the algebraic expression \(4 + y\). This expression shows a simple summation involving an unknown variable \(y\) and a known number 4. The beauty of algebraic expressions is that they can model real-world situations where some values are unknown, and through various algebraic techniques, these values can be discovered.

Transposition in Equations

Transposition is a strategy used in algebra to isolate the variable and solve equations. By 'moving' a term from one side of the equation to the other, we can simplify the equation and make the variable easier to solve for. The key is to perform the transposition while keeping the equation balanced through inverse operations.

In our exercise, transposing +4 to the other side involves turning it into -4, based on the notion that adding 4 and subtracting 4 cancel each other out (they are inverse operations). This is shown in the step: \(y = 12 - 4\). It's crucial to remember that whatever operation you perform on one side of the equation must also be performed on the other side to maintain equality. Transposition is a valuable tool in solving not just simple, but also complex equations.

Equation Evaluation

Once an equation is simplified through techniques like transposition, the next step is to evaluate it, which means performing the arithmetic operations to find the value of the variable. This is essentially where we calculate the answer based on the operations defined in the simplified equation.

For our exercise, after transposition, the evaluation involved the simple arithmetic of subtracting 4 from 12, which gives us \(y = 8\). Evaluation is straightforward in simple equations like this, but it's a critical process in algebra: getting to the point where you can perform this final calculation means you've successfully maneuvered through the equation. Evaluating the simplified equation results in the solution that 'y' must be 8 in order for the original algebraic expression to be true.