Problem 30 - Algebra 1

Question

\(\frac{1}{2}+t\) when \(t=\frac{1}{2}\)

Step-by-Step Solution

Answer

The solution for the equation \( \frac{1}{2}+t \) when \( t=\frac{1}{2} \) is 1

Step 1: Understand the Equation

We have an equation which is \(\frac{1}{2} + t\), where \( t \) is a variable which we can substitute with a given value.

Step 2: Substitute \( t \)

We replace \( t \) in the equation with the given value, which is \(\frac{1}{2}\). So our equation becomes \(\frac{1}{2} + \frac{1}{2}\).

Step 3: Perform the arithmetic

Adding the two fractions together gives \( \frac{1}{2} + \frac{1}{2} = 1 \)

Key Concepts

Addition of Fractions

Understanding how to add fractions is an essential skill in basic arithmetic. To add fractions, it's important to ensure that they have the same denominator, which is the bottom part of the fraction. In this exercise, we have two fractions: \(\frac{1}{2}\) and \(t\), which is also equal to \(\frac{1}{2}\) after substitution.

Here's how you add fractions with the same denominators:

• The numerators, which are the numbers on top of the fractions, are added directly. So in our case, that means \(1 + 1 = 2\).
• You keep the denominator the same. Since we started with denominators of 2, our resulting fraction is \(\frac{2}{2}\).
• \(\frac{2}{2}\) simplifies to 1 because 2 is half of 2.

Remember this basic concept: when adding fractions with the same denominator, only add the numerators. If they have different denominators, they need to be adjusted first.

Substitution

Substitution in algebra involves replacing a variable with a given number to simplify an expression or solve an equation. It's like finding a missing puzzle piece and putting it into place. In the given exercise, we had an algebraic expression \(\frac{1}{2} + t\). The next step was to replace the variable \(t\) with a provided value.

Here's a quick guide on substitution:

• Identify the variable you need to substitute. Here, \(t\) is our variable.
• Insert the value of the variable into the equation. In this case, \(t\) was replaced by \(\frac{1}{2}\).
• Simplify the expression with the new value in place to find your answer.

Substitution is a fundamental skill in algebra and helps transform abstract expressions into solvable math problems.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition and multiplication). They are used to represent mathematical relationships. In this exercise, the expression \(\frac{1}{2} + t\) contains a fixed number and a variable.

Key points about algebraic expressions:

• Variables are symbols that can stand for different numbers. They add flexibility to expressions.
• Constants are fixed numbers. In our example, \(\frac{1}{2}\) is a constant.
• Operations define how the variables and constants interact, such as addition in this case.

Understanding algebraic expressions allows you to work through various math problems by translating real-world scenarios into mathematical terms. Once variables are substituted with specific values, you can perform arithmetic to solve them.