Problem 63 - Algebra 1: Concepts and Skills

Question

Solve the equation. \(8+y=3\)

Step-by-Step Solution

Answer

The value of y in the given equation is -5.

Step 1: Rearrange the equation to isolate y

The initial equation is \(8 + y = 3\). In order to find the value of y, subtract 8 from both sides of the equation so that y is alone on one side. This gives \(y = 3 - 8\).

Step 2: Simplify the right side of the equation

Once the equation is rearranged, simplify the right side of the equation to find the solution. This would look like: \(y = -5\).

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is fundamental to solving linear equations. An algebraic expression is a combination of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division.

For instance, in the equation 8 + y = 3, 8 + y constitutes an algebraic expression with y as the variable. It's important for students to familiarize themselves with recognizing these expressions and the operations within them to be adept at algebraic manipulation.

When approaching such expressions in homework, take the time to identify the parts: constants like 8, variables like y, and the operation, in this case, addition. Then, follow the algebraic rules to manipulate these expressions to solve for the unknown variable.

Isolate Variable

The technique of isolating the variable is a crucial step in solving linear equations. To isolate the variable means to get the variable on one side of the equation by itself. Doing so helps us find its value.

In the given exercise 8 + y = 3, we want to find out what y equals. Therefore, we perform operations that 'move' all other numbers to the opposite side of the equation from y. To isolate y, we subtract 8 from both sides, which cancels the 8 on the left and moves it to the right.

Remember, whatever operation you do to one side of the equation, you must do to the other to maintain the equality. This principle will guide you when you're trying to isolate the variable, whether by adding, subtracting, multiplying, or dividing both sides by the same number.

Equation Simplification

Once you have isolated the variable, equation simplification is the next step. This process involves reducing the equation to its simplest form to find the solution.

In our exercise, after isolating the variable we get y = 3 - 8. Now, we simply perform the subtraction: 3 - 8 equals -5. Hence, y = -5 is the simplified solution of the equation. It's crucial to simplify correctly to find the accurate value of the isolated variable.

While simplifying an equation, combining like terms, and performing basic arithmetic correctly, are key aspects. Students can encounter mistakes if not careful with signs (positive/negative) and arithmetic operations. Always double-check your work to ensure accuracy in the simplified solution, leading you to the correct answer.