Problem 14 - Prealgebra

Question

Solve each equation. Check your solution. $$8 n+3=-5$$

Step-by-Step Solution

Answer

The solution to the equation is \(n = -1\).

Step 1: Isolate the Variable

To solve for \(n\), we need to isolate it on one side of the equation. We start by subtracting 3 from both sides of the equation to remove the constant term from the left side. This gives us:\[8n + 3 - 3 = -5 - 3\]Simplifying both sides, we get:\[8n = -8\]

Step 2: Solve for \(n\)

Now that we have \(8n = -8\), we need to solve for \(n\) by dividing both sides of the equation by 8:\[\frac{8n}{8} = \frac{-8}{8}\]This simplifies to:\[n = -1\]

Step 3: Check the Solution

To ensure that \(n = -1\) is correct, we substitute \(n = -1\) back into the original equation \(8n + 3 = -5\) and check if both sides are equal:\[8(-1) + 3 = -5\]Calculate each term:\[-8 + 3 = -5\]Both sides indeed equal \(-5\), confirming our solution is correct.

Key Concepts

Isolate the Variable

When solving linear equations, one of the essential initial steps is to isolate the variable. This means we want to have the variable on one side of the equation by itself and all the other terms on the other side. By isolating the variable, we can clearly see what value the variable holds.In our example, we started with the equation \(8n + 3 = -5\). To isolate \(n\), we need to remove the constant from the same side as \(n\). We do this by performing the opposite operation of addition, which is subtraction. So, we subtract 3 from both sides:- Perform \(8n + 3 - 3 = -5 - 3\).- Simplifying gives us \(8n = -8\).By rewriting the equation in terms of \(8n = -8\), we've successfully isolated \(n\) on one side. This simplifies the process of solving the equation.

Checking Solutions

Checking your solution is a vital step to ensure the accuracy of your results when solving equations. It helps verify that the solution indeed satisfies the original equation.Once we obtained \(n = -1\), we substituted this value back into the original equation to check our work:- Substitute \(n = -1\) into \(8n + 3 = -5\): - Compute \(8(-1) + 3\).- Calculate: - This simplifies to \(-8 + 3\), which equals \(-5\).Both sides of the equation are equal, confirming that our solution is correct. Always substituting back and calculating ensures that the error hasn't slipped through during simplification or arithmetic processing.

Equation Simplification

Equation simplification is a fundamental step that involves reducing the equation to its simplest form. It makes it easier to isolate and solve for variables.In our case, after subtracting 3 from both sides, we reached the simpler equation \(8n = -8\). From there, we performed division by 8:- Perform \(\frac{8n}{8} = \frac{-8}{8}\).- Simplify to obtain \(n = -1\).Simplification also involves consistently applying arithmetic operations, such as adding, subtracting, multiplying, or dividing evenly on both sides of the equation. This helps keep the equation balanced, allowing us to isolate variables and solve them straightforwardly. By focusing on simplification, you make the path to finding the solution smoother and ensure your solution is correct by maintaining equality.