Problem 35 - Prealgebra

Question

Using the addition property of equality first, solve each of the following equations. $$-11 a+4=-29$$

Step-by-Step Solution

Answer

The solution is \(a = 3\).

Step 1: Identify the equation

The given equation is \(-11a + 4 = -29\). Our goal is to find the value of \(a\).

Step 2: Isolate terms with variable

We need to start by isolating the term with \(a\). Subtract 4 from both sides of the equation. This step uses the addition property of equality.\[-11a + 4 - 4 = -29 - 4\]Simplifying both sides, we get:\[-11a = -33\]

Step 3: Solve for the variable

Now, divide both sides by -11 to solve for \(a\):\[a = \frac{-33}{-11}\]This simplifies to:\[a = 3\]

Step 4: Verify the solution

To verify, substitute \(a = 3\) back into the original equation:\(-11(3) + 4 = -29\)\(-33 + 4 = -29\)Simplifying gives \(-29 = -29\), confirming our solution.

Key Concepts

Solving Equations

Equations are like puzzles that help us discover unknown values. Solving an equation means finding the value of the variable that makes the equation true. In the equation \(-11a + 4 = -29\), our task is to solve for the variable \(a\).
To solve the equation, we need to make sure that the equation is balanced after each step. This means that whatever we do to one side of the equation, we must also do to the other side.
In our initial step, we identified the need to eliminate the constant term \(+4\) so that we can focus on isolating \(a\). By subtracting 4 from both sides of the equation, we keep the equation balanced and move a step closer to solving for \(a\).
Overall, the goal is to use basic arithmetic operations to simplify the equation until the variable is isolated, allowing us to find its value.

Prealgebra

Prealgebra is a foundational level of mathematics that introduces basic algebraic concepts. These concepts prepare students for more complex algebra studies.
In prealgebra, students learn about variables, expressions, and equations. A variable is a symbol, often a letter, that represents an unknown number. In our exercise, \(a\) is the variable.
Prealgebra covers techniques like the addition property of equality. This property is a rule that states you can add or subtract the same number from both sides of an equation without changing its solution.
By understanding and applying these concepts, students gain the tools needed to solve equations in a logical and methodical manner. Prealgebra is essential for building the skills that enable students to tackle increasingly challenging math problems.

Variable Isolation

Variable isolation is a key step in solving equations. It involves rearranging the equation so that the variable stands alone on one side. This makes it easier to identify its value.
The process starts by moving constants away from the variable. In our example, we isolated \(-11a\) by subtracting 4 from both sides:

• This left us with \(-11a = -33\).

Next, we focused on removing the coefficient of the variable. We achieved this by dividing both sides by \(-11\) to solve for \(a\):

• This operation gives us \(a = 3\).

By isolating the variable, we effectively unlock the solution to the exercise. It's crucial to handle each step with care, ensuring that each move maintains the equation's balance.