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Chapter 0: Problem 3

Solve for the indicated variable. $$5 t+11=18$$

Short Answer

Expert verified
\( t = \frac{7}{5} \) or \( t = 1.4 \).

Step by step solution

01

Identify the Equation

The equation we are working with is \( 5t + 11 = 18 \). We need to find the value of \( t \).
02

Isolate the Term with the Variable

To isolate the term with \( t \), subtract 11 from both sides of the equation: \( 5t + 11 - 11 = 18 - 11 \). Simplifying both sides gives us \( 5t = 7 \).
03

Solve for the Variable

Divide both sides by 5 to solve for \( t \): \( \frac{5t}{5} = \frac{7}{5} \). Thus, \( t = \frac{7}{5} \) or \( t = 1.4 \).

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Key Concepts

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

Isolating Variables
When solving linear equations, the goal is often to find the value of a variable. This process is known as isolating the variable, where we rearrange the equation so that the variable is by itself on one side of the equation. In our original exercise, we're tasked with finding the value of \( t \) in the equation \( 5t + 11 = 18 \). To isolate \( t \), we must first isolate the term containing \( t \). This means getting \( 5t \) by itself before further simplifying.To begin:
  • Look at the equation \( 5t + 11 = 18 \).
  • The term \( 5t \) is currently added to 11.
  • We counteract this by subtracting 11 from both sides, so \( 5t + 11 - 11 = 18 - 11 \), simplifying to \( 5t = 7 \).
This leaves us one step closer to uncovering \( t \). Isolating the variable always involves eliminating any constants and coefficients keeping it from standing alone.
Solving Equations
After isolating the term with the variable, the next step is solving the equation, which means finding the actual numerical value of the variable. In the case of a simple linear equation like \( 5t = 7 \), we can directly solve for \( t \) by performing the inverse operation of multiplication, which is division.Here's how you continue:
  • The equation is currently \( 5t = 7 \).
  • Since \( t \) is multiplied by 5, divide both sides by 5 to solve for \( t \).
  • This gives \( \frac{5t}{5} = \frac{7}{5} \), simplifying to \( t = \frac{7}{5} \) or \( t = 1.4 \).
By dividing both sides by 5, we have effectively solved the equation and isolated \( t \) to find its correct value of 1.4.
Algebraic Manipulation
Algebraic manipulation involves using various algebraic techniques to simplify and solve equations. These techniques often include adding, subtracting, multiplying, or dividing both sides of an equation by the same number to maintain equality.In our problem:
  • We subtracted 11 from both sides to eliminate the constant added to \( 5t \).
  • After obtaining \( 5t = 7 \), we divided both sides by 5 to simplify and solve for \( t \).
Each step follows the fundamental principle of maintaining balance in an equation. This means whatever you do to one side, you must do to the other. This makes sure the equation remains true. Algebraic manipulation is crucial for transforming complex problems into simpler ones, allowing for straightforward solutions.

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