Problem 31 - Prealgebra

Question

Solve each equation. Check your solution. $$8-t=-25$$

Step-by-Step Solution

Answer

The solution to the equation is \( t = 33 \).

Step 1: Isolate the variable

To solve the equation, we need to isolate the variable \( t \). Start by subtracting 8 from both sides of the equation to get rid of the constant term on the left side:\[-t = -25 - 8\]This simplifies to:\[-t = -33\]

Step 2: Solve for the variable

Now that the variable \( t \) is isolated with its coefficient, we need to solve for \( t \). The coefficient of \( t \) is -1, so multiply both sides of the equation by -1 to solve for \( t \):\[t = 33\]

Step 3: Check the solution

To ensure our solution is correct, substitute \( t = 33 \) back into the original equation:\[8 - t = -25\]Substitute \( t \) with 33:\[8 - 33 = -25\]Since both sides of the equation are equal, our solution \( t = 33 \) is correct.

Key Concepts

Checking Solutions

"Checking solutions" is a vital step to make sure the answer we found is truly correct. When solving linear equations, it's essential to verify your solution by plugging it back into the original equation. If both sides equal after substitution, it confirms that your solution is accurate.
Substituting the solution back into the equation acts as a double-check against errors in computation. In our exercise, with the solution found as \( t = 33 \), we substitute it back into the original equation:

• Original equation: \(8 - t = -25\)
• Substitute: \(8 - 33 = -25\)

The expressions on both sides of the equation simplify to \(-25\). Since they match, it confirms that the calculated value for \( t \) is correct. Always remember to verify your answers in this way to ensure no steps were missed or errors made during calculations.

Isolation of Variables

"Isolation of variables" is the method where you rearrange an equation to solve specifically for one variable, typically the one you're interested in. In a linear equation, the goal is to have the variable on one side and everything else on the other.
This process helps make the solution visible. Consider this equation from our problem:

• Starting equation: \(8 - t = -25\)
• Subtract 8 from both sides: \(-t = -25 - 8\)
• Simplify: \(-t = -33\)

Now, the variable \( t \) stands alone on one side. This clear separation allows us to directly focus on finding its value. Isolation of variables sets the stage for effortless calculation, making it a crucial skill in algebra.

Equation Balancing

"Equation balancing" is all about maintaining equality by doing equivalent operations on both sides of an equation. This technique ensures the integrity of the equation is preserved, and both sides remain equal through the process.
In the given exercise, we began by subtracting 8 from both sides:

• This helps eliminate additional terms and focuses solely on the variable \( t \).
• After simplification, \(-t = -33\), we solved for \( t \) by multiplying both sides by -1.

These actions kept the balance intact right through to reaching the solution, \( t = 33 \). Continuing to balance equations in every step is what successfully leads us to an accurate solution, making it a fundamental principle in solving equations. By understanding and applying this concept well, students can confidently tackle any linear equation they encounter.