Problem 43 - Algebra 1: Concepts and Skills

Question

Solve the equation by simplifying first. $$ t-(-4)=4 $$

Step-by-Step Solution

Answer

The solution to the equation is \( t=0 \).

Step 1: Simplify the Equation

The first step is to simplify the equation. To do this, recognize that the double negative '-(-4)' can be turned into a positive. So, the original equation \( t-(-4)=4 \) simplifies to \( t+4=4 \).

Step 2: Solve for 't'

Next, isolate 't' by subtracting 4 from both sides of the equation to keep the equation balanced. This simplification will give the final equation: \( t+4-4=4-4 \) which simplifies to \( t=0 \).

Key Concepts

Simplifying Expressions

Simplifying expressions is a foundational skill in algebra that helps us to manage and solve equations efficiently. It involves reducing expressions to their simplest form by performing operations like addition, subtraction, multiplication, division, and removing parentheses.

When you encounter an algebraic expression with multiple terms and operations, it's crucial to follow the proper order of operations—often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Through simplification, we reduce the complexity of the problem and prevent mistakes in further calculations.

In our exercise, the expression within the equation involves a double negative, '-(-4)', which according to arithmetic rules, converts to a positive '+4'. This is because two negatives cancel out, making a positive. By simplifying the expression, we transform the initial equation from \( t-(-4)=4 \) into a simpler one: \( t+4=4 \).

Isolating Variables

Isolating variables is the process of rearranging an algebraic equation to make one variable the subject of the equation. This means moving all other terms to the opposite side of the equation from the variable you are solving for.

The goal is to have the variable alone on one side of the equation, effectively 'isolating' it. To isolate a variable, you perform the inverse operation to the terms surrounding it and do the same to the other side of the equation to maintain balance, as dictated by the properties of equality.

For example, in the exercise \( t+4=4 \), we isolate 't' by subtracting '4' from both sides. We thus perform the inverse of addition, which is subtraction, giving us the new equation \( t+4-4=4-4 \), which simplifies further to \( t=0 \). It’s crucial to do the same operation on both sides to keep the equation equal; this is where the concept of balance in equations plays a vital role.

Properties of Equality

Properties of equality are a set of rules that allow us to manipulate equations while maintaining their balance and integrity. These properties state that if you perform the same operation on both sides of an equation, the two sides remain equal.

Some of the key properties include the Addition Property of Equality (if \( a = b \) then \( a+c = b+c \) for any number 'c'), and the Subtraction Property of Equality (if \( a = b \) then \( a-c = b-c \) for any number 'c'). Multiplication and Division Properties follow similar logic.

In our example, when we simplified \( t+4-4=4-4 \), we applied the Subtraction Property of Equality to subtract '4' from both sides, ensuring the two sides remained equal. Using these properties carefully is key to correctly solving algebraic equations.