Problem 45 - Algebra 1: Concepts and Skills

Question

SOLVING WITH MENTAL MATH Use mental math to solve the equation. $$ 3+y=8 $$

Step-by-Step Solution

Answer

The solution to the equation is \(y = 5\).

Step 1: Rearrange the equation

In this case, you start by rearranging the equation to solve for y. This involves subtracting 3 from both sides of the equation: \(3 + y = 8\) becomes \(y = 8 - 3\).

Step 2: Perform the calculation

Next, you subtract 3 from 8 to find the value of y: \(y = 8 - 3 = 5\).

Key Concepts

Solving Equations

Solving equations is the process of finding the value of an unknown variable that makes the equation true. It's like a puzzle where you balance both sides to find the missing piece. In this exercise, we're trying to find the value of \( y \) in the equation \( 3 + y = 8 \). Solving it involves performing operations that will isolate \( y \) on one side.

• The equation is linear, meaning there is no variable raised to a power or inside a square root.
• Start by identifying the terms containing the variable and constants on either side of the equal sign.
• Decide what operation will best help to isolate the variable. In this case, subtract 3 from both sides.

By using such steps, you effectively "unpack" the equation to reveal the solution, similar to peeling layers from an onion. Calculating \( y = 8 - 3 \), we find that \( y = 5 \).

Mathematical Reasoning

Mathematical reasoning enables you to think logically and determine problem-solving strategies. It’s essential for understanding not just how to solve an equation, but why each step is necessary.

• In the given equation \(3 + y = 8\), the reasoning helps us understand that any operation done on one side must be done to the other to keep the equation balanced.
• This is based on the principle of equality, which maintains that if you perform the same operation on both sides of an equation, the equality still holds true.
• Reasoning ensures that as we subtract 3 from the left to cancel it out, subtracting 3 from the right maintains the balance.

Using this logical approach helps reinforce why manipulating an equation in certain ways leads to finding the correct answer. It turns abstract numbers into concrete understanding.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to simplify expressions or find values of variables. It's a fundamental skill in algebra, used to make complex problems easier to solve.

• The initial step in the solution \(3 + y = 8\) exemplifies basic algebraic manipulation: subtracting 3 from both sides to isolate \( y \).
• Such manipulation can involve addition, subtraction, multiplication, or division.
• Its goal is to transform the equation into a form where the value of the variable is clear.

For this equation, the manipulation required is straightforward, focusing on moving numbers from one side to the other. Mastering these techniques makes handling more complex algebraic problems far less daunting.