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Chapter 2: Problem 43

Solve each equation and check the result. $$ t=\frac{7 t-9}{16} $$

Short Answer

Expert verified
The solution is \( t = -1 \).

Step by step solution

01

Eliminate the Fraction

To solve the equation \( t = \frac{7t - 9}{16} \), start by eliminating the fraction. Multiply both sides of the equation by 16 to get rid of the denominator: \[16t = 7t - 9\]
02

Isolate the Variable Term

Next, isolate the term with \( t \) on one side. Subtract \( 7t \) from both sides:\[16t - 7t = -9\]which simplifies to:\[9t = -9\]
03

Solve for the Variable

Now, solve for \( t \) by dividing both sides of the equation by 9:\[t = \frac{-9}{9}\]This simplifies to:\[t = -1\]
04

Check the Solution

Substitute \( t = -1 \) back into the original equation to verify:\[t = \frac{7(-1) - 9}{16}\]Simplifying inside the fraction:\[\frac{-7 - 9}{16} = \frac{-16}{16} = -1\]The left side \( t = -1 \) matches the solved value.

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Key Concepts

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

Eliminating Fractions
When solving linear equations, it's often useful to eliminate any fractions. This keeps things simpler and more straightforward. To eliminate a fraction, multiply every term in the equation by the denominator of the fraction you wish to remove. Let's take a closer look at how to apply this.
Consider the equation \( t = \frac{7t - 9}{16} \). The fraction here has a denominator of 16. To eliminate it, multiply both sides of the equation by 16:
  • Left side: \( 16 \times t = 16t \)
  • Right side: \( 16 \times \frac{7t - 9}{16} = 7t - 9 \)
This multiplication effectively removes the fraction by canceling out the 16 in the denominator.
Now, you have a simpler equation to work with, \( 16t = 7t - 9 \), which is much easier to solve.
Isolating Variables
After eliminating fractions, the next goal is to isolate the variable. Isolation means getting the variable, in this case, \( t \), by itself on one side of the equation. Let's continue from where we left off.
The equation \( 16t = 7t - 9 \) needs to have \( t \) isolated. To do this, subtract \( 7t \) from both sides:
  • Starting equation: \( 16t = 7t - 9 \)
  • Subtract \( 7t \) from both sides: \( 16t - 7t = -9 \)
Now, simplify that by combining like terms:
  • Combine \( 16t - 7t \) to get \( 9t \)
You are left with the equation \( 9t = -9 \). Notice that \( t \) is almost isolated, needing just one more step.
Verifying Solutions
Once you find a value for the variable, it's critical to verify that your solution satisfies the original equation. Verifying ensures that no mistakes were made during the process.
Let's use \( t = -1 \) to check our work for the equation \( t = \frac{7t - 9}{16} \). Substitute \( t = -1 \) into the original equation:
  • Left side: \( t = -1 \)
  • Right side: \( \frac{7(-1) - 9}{16} \)
  • Simplify the right side: \( \frac{-7 - 9}{16} = \frac{-16}{16} = -1 \)
The right side equals \( -1 \), which matches the left side. This confirms that \( t = -1 \) is indeed the correct solution.
Verification helps solidify understanding and provides confidence that the problem was solved correctly.

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