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Chapter 2: Problem 114

Solve each equation requiring simplification. $$-18-7=5 t-9 t-6 t$$

Short Answer

Expert verified
The solution is \( t = 2.5 \).

Step by step solution

01

Simplify the Right Side

Combine like terms on the right-hand side of the equation. \[ 5t - 9t - 6t = (5 - 9 - 6)t = -10t \]Now the equation looks like: \[ -18 - 7 = -10t \]
02

Simplify the Left Side

Combine the constants on the left-hand side of the equation. \[ -18 - 7 = -25 \]Now the equation is simplified to: \[ -25 = -10t \]
03

Solve for t

Isolate the variable t by dividing both sides of the equation by -10. \[ t = \frac{-25}{-10} = 2.5 \]

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Key Concepts

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

Simplifying Expressions
One of the first steps in solving algebraic equations is simplifying expressions. This means making the equation more manageable and easier to solve by reducing it to its simplest form. Simplifying might include:
  • Combining like terms.
  • Breaking down complex fractions.
  • Factoring expressions.
In our example, we first simplify the right side by combining like terms which involves adding and subtracting the coefficients of the terms. This is crucial as it lays the groundwork for further steps and ensures accuracy in calculations.
Combining Like Terms
Combining like terms is an essential process in algebra. It involves adding or subtracting coefficients of terms that have the same variable raised to the same power. For instance, consider the equation we had:
\(5t - 9t - 6t\)
To combine these terms effectively, you add or subtract the coefficients while keeping the variable and its power intact:
\[ 5 - 9 - 6 = -10 \]
Thus, simplifying \(5t - 9t - 6t\) results in \(-10t\). Combining like terms helps in reducing the complexity of the equation, making it simpler to solve. Always look for like terms on both sides of the equation and combine them as early as possible.
Isolating Variables
After simplifying expressions and combining like terms, the next pivotal step is isolating the variable. Isolating the variable means getting the variable you are solving for alone on one side of the equation. Here's how we do it:
In the simplified equation:
\[-25 = -10t\]
We need to get \(t\) by itself. To do that, we divide both sides of the equation by the coefficient of \(t\), which is \(-10\). This gives us:
\[\frac{-25}{-10} = t = 2.5 \]
This step is crucial as it provides the final solution to the equation. Isolating the variable ensures we accurately determine the value of the variable we are solving for.

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