Problem 30 - Algebra 1

Question

Solve the equation. $$t-(-4)=4$$

Step-by-Step Solution

Answer

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

Step 1: Simplify the subtraction

We first simplify the expression \(t-(-4)\) to \(t+4\), as subtracting a negative number is equivalent to adding the positive number.

Step 2: Solve for variable t

To solve for \(t\), subtract 4 from both sides of the equation. The equation becomes \(t+4-4=4-4\), which simplifies to \(t=0\).

Key Concepts

Equation Simplification

Solving equations often starts with the process known as equation simplification. This is an essential part of algebra since it helps to transform complex equations into simpler forms that are easier to work with. Simplification may involve combining like terms, using the distributive property, or converting subtraction of a negative number into addition of a positive one.

For example, in the exercise given, you begin with the equation \(t-(-4)=4\). To simplify it, you need to recognize that subtracting a negative is the same as adding its positive counterpart. Thus, the simplification process changes \(t-(-4)\) to \(t+4\). This transformation is pivotal as it sets the stage for the next steps in solving the equation.

Subtracting Negative Numbers

Many students find subtracting negative numbers tricky, but a simple rule can clarify this concept: when you subtract a negative, you can think of it as adding its opposite, the positive version of the number. This is because negative and positive numbers are on opposite sides of zero on the number line, and subtracting a number is the same as moving left; if that number is negative, you're moving left of a left position, which actually ends up going right.

In our exercise, subtracting -4 is the same as adding 4. Remember that two negatives make a positive, so \(t - (-4)\) becomes \(t + 4\). This rule simplifies equations and ensures that you're always moving in the right direction when managing negative numbers.

Solving for Variables

The end goal in an algebraic equation is often to find the value of the variable. This means you want to isolate the variable on one side of the equation. You do this through various operations like addition, subtraction, multiplication, or division, applied inversely to both sides of the equation to keep it balanced.

In the given exercise, once the equation is simplified to \(t + 4 = 4\), the next step is to isolate \(t\). By subtracting 4 from both sides of the equation, you're using the inverse operation of the addition that's currently affecting \(t\). After doing that, you'll arrive at \(t = 0\), which means you've found the value for \(t\). With practice, solving for variables becomes a systematic process that can be applied to more complex and real-world problems.