Problem 33 - Prealgebra

Question

Solve each equation. Check your solution. $$8=-5-b$$

Step-by-Step Solution

Answer

The solution to the equation is \(b = -13\).

Step 1: Isolate the variable

The given equation is \(8 = -5 - b\). To isolate \(b\), add \(5\) to both sides of the equation. The equation becomes: \[8 + 5 = -5 + 5 - b\] which simplifies to: \[13 = -b\].

Step 2: Solve for \(b\)

Now, to solve for \(b\), multiply both sides by 1 to get: \[-13 = b\] or equivalently: \[b = -13\].

Step 3: Verify the solution

Substitute \(b = -13\) back into the original equation to verify:Original equation: \(8 = -5 - b\)Substitute \(-13\) for \(b\):\(8 = -5 - (-13)\)This simplifies to:\(8 = -5 + 13\) which equals \(8 = 8\). The solution is verified as correct.

Key Concepts

Solving Equations

Understanding how to solve equations is fundamental in prealgebra. In simpler terms, solving equations means finding the value of the unknown variable that makes the equation true. For the given exercise, you have the equation \(8 = -5 - b\). The goal here is to determine what value of \(b\) makes both sides of the equation equal.

The process typically involves a few systematic steps. First, you want to simplify the equation by combining like terms on either side if possible. However, in this example, the right side of the equation contains a number and a variable that cannot be combined directly without further steps.

Simplifying equations can make them easier to solve, as they reduce clutter and set the stage for isolating the variable effectively.

Isolate Variable

To successfully solve any algebraic equation, one must learn how to isolate the variable. When we say "isolate the variable," we mean rearranging the equation so the variable stands alone on one side of the equation.

In the example \(8 = -5 - b\), our task is to get \(b\) by itself. Helpful steps include performing operations that cancel out numbers on the side of the variable.

• First, add \(5\) to both sides to neutralize the \(-5\) on the right side, turning the equation into \(13 = -b\).
• Finally, you deal with the negative sign in front of \(b\) by multiplying or dividing both sides by \(-1\), resulting in \(b = -13\).

These actions ensure the variable is isolated, allowing us to directly see its value.

Verify Solution

After isolating the variable and solving for it, verifying the solution is the crucial final step. This process ensures the solution you found is indeed correct.

To verify, substitute the value of the variable back into the original equation and check if the equation holds true.

• Take your solution \(b = -13\) and substitute it into the original equation \(8 = -5 - b\).
• Replace \(b\) with \(-13\), converting the equation to \(8 = -5 - (-13)\).
• When simplified, \(8 = -5 + 13\), which further simplifies to \(8 = 8\).

Since both sides of the equation equal each other, your solution is verified as accurate. Verifying assures that no mistakes were made and confirms the reliability of your solution.