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Chapter 3: Problem 7

Solve each equation. First combine any like terms on each side of the equation. \(24=t+3 t\)

Short Answer

Expert verified
The solution to the equation is \(t = 6\).

Step by step solution

01

Combine Like Terms

The given equation is \( 24 = t + 3t \). Identify and combine like terms on the right side of the equation. Both \(t\) and \(3t\) are like terms since they both involve \(t\). By combining these, we obtain \(4t\). The equation becomes \(24 = 4t\).
02

Isolate the Variable

To solve for \(t\), we need to isolate it on one side of the equation. Divide both sides of the equation by 4 to achieve this. The equation \(24 = 4t\) becomes \( \frac{24}{4} = \frac{4t}{4} \).
03

Simplify Both Sides

Simplify both sides of the equation from Step 2. \( \frac{24}{4} \) simplifies to 6, and \( \frac{4t}{4} \) simplifies to \(t\). Therefore, we have \(t = 6\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Equations
Solving an equation is like finding the answer to a mystery. We want to figure out what number a particular variable represents. An equation is a mathematical statement that shows the equality between two expressions.

For example, in the equation \(24 = t + 3t\), the goal is to find the value for \(t\) that makes this equation true. The key steps in solving equations involve:
  • Understanding what the equation is asking.
  • Performing operations to simplify the equation if needed.
  • Applying techniques like combining terms or isolating the variable to find the solution.
This structured approach helps to systematically reduce the equation until you solve for the variable in question.
Combining Like Terms
In algebra, "like terms" refer to terms that have the same variables raised to the same power. They can be combined to simplify an equation. This is an essential step in reducing the complexity of algebraic expressions.

In the equation \(24 = t + 3t\), both \(t\) and \(3t\) are like terms because they contain the variable \(t\). You can sum them up as they share the same variable. Here, adding \(t + 3t\) gives \(4t\), simplifying the equation to \(24 = 4t\).

Combining like terms is crucial because it:
  • Helps reduce the equation to its simplest form.
  • Allows clearer insight into the relationships between variables and constants.
  • Makes it easier to perform further operations to solve the equation.
Always look for like terms on the same side of the equation first before proceeding with solving the equation.
Isolating Variables
Once you have simplified an equation by combining like terms, the next step is to isolate the variable. This means getting the variable all alone on one side of the equation, keeping it simple to solve.

Taking our example, after combining like terms we have \(24 = 4t\). Isolating the variable \(t\) requires removing the coefficient that is multiplying it. Division is a handy tool here. Divide every term in the equation by 4, the coefficient of \(t\), resulting in \(\frac{24}{4} = \frac{4t}{4}\).

Simplify this to get \(6 = t\), or more commonly \(t = 6\).
  • Isolating variables clarifies what value the variable must have to balance the equation.
  • It involves using inverse operations such as addition/subtraction and multiplication/division.
  • This step is important in pinpointing the exact solution to an algebra problem.
Mastering isolating variables is crucial for solving almost any algebraic equation efficiently.

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Most popular questions from this chapter

Solve each equation. $$9 a+29+7=0$$

Write each phrase as a variable expression. Use \(x\) to represent "a number." The sum of -7 and a number

Solve each equation. $$-8 n+1=-6 n-5$$

Write each sentence as an equation. The product of -5 and -29 gives \(145 .\)

Is the equation \(-x=6\) solved for the variable? Explain why or why not.
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