Problem 81 - Algebra 1

Question

Solve the equation. $$-9-(-a)=-2$$

Step-by-Step Solution

Answer

The solution to the equation \(-9-(-a)=-2\) is \(a = 7\).

Step 1: Simplification

The equation given is \(-9-(-a)=-2\). The first step is to simplify. When you see -- in a math equation, it can be simplified to '+'. So, \(-9-(-a)\) simplifies to \(-9+a\). So the equation now is \(-9 + a = -2\).

Step 2: Isolate variable a

To isolate \(a\), add 9 to both sides of the equation. This will give \(a = -2 + 9\).

Step 3: Solve for a

Summing \(-2\) and \(9\) gives \(7\), so the solution to the equation is \(a = 7\).

Key Concepts

Simplification

Simplification in mathematics is all about making expressions easier to understand and manage.
It often involves reducing the complexity of the expression or equation. In our exercise, we began with the equation:
\[-9-(-a)=-2\]When you see a double negative sign, like -(-something), it translates to a positive. This is because subtracting a negative is identical to adding a positive. So, in the equation, \(-9-(-a)\) becomes \(-9+a\). Now, the equation looks simpler as:
\[-9 + a = -2\]The purpose of simplification is to allow us to more clearly see how to proceed with solving the equation. By reducing clutter, we can focus on the steps needed to solve for the unknown variable. Remember, simplification doesn't change the value or outcome—it only makes it easier to solve.

Isolation of Variables

The next vital step in solving equations is the isolation of the variable.
Isolation means manipulating the equation in a way that leaves the variable we are solving for—here, it's \(a\)—by itself on one side of the equation. This is essential because solving for a variable translates to finding its particular value.
In our example, after simplifying, we have:
\[-9 + a = -2\]To isolate \(a\), we need to remove the \(-9\) from the left side. This is done by performing the opposite operation. So if \(-9\) is subtracted, the opposite operation is adding \(9\). This action helps us get a clear expression for \(a\) on one side, progressing us towards a solution:\[a = -2 + 9\]Remember, the goal of isolation is to have one instance of the variable by itself, which directly allows us to compute its value.

Addition Property of Equality

The addition property of equality is a fundamental principle used to keep equations balanced while solving them. It states that adding the same number to both sides of an equation does not affect the equality.
Think of an equation like a balanced scale. If you add something to one side, you must add the equal amount to the other side to maintain balance.
For instance, in our equation:
\[-9 + a = -2\]We needed to get rid of \(-9\) to isolate \(a\). By adding \(9\) to both sides, we maintain balance:\[(-9 + 9) + a = -2 + 9\]This simplifies to:\[a = 7\]Using the addition property of equality ensured that we kept the equation true while rearranging it. It’s a reliable tool in math to handle variables and solve equations effectively.