Problem 25 - Prealgebra

Question

Using the addition property of equality first, solve each of the following equations. $$6 x-5=19$$

Step-by-Step Solution

Answer

The solution is \(x = 4\).

Step 1: Isolate the Variable Term

Begin by using the addition property of equality to eliminate the constant term on the left-hand side of the equation. Add 5 to both sides of the equation.\[6x - 5 + 5 = 19 + 5\]This simplifies to:\[6x = 24\]

Step 2: Solve for the Variable

Now that we have isolated the variable term, divide both sides of the equation by 6 to solve for \(x\).\[\frac{6x}{6} = \frac{24}{6}\]This simplifies to:\[x = 4\]

Key Concepts

Solving Equations

Solving equations is a fundamental skill in mathematics, crucial for tackling problems in prealgebra and beyond. The goal of solving an equation is to find the value of the unknown variable that makes the equation true. In the given exercise, we start with the equation \(6x - 5 = 19\). To solve it:

• First, understand that the equation expresses a balance between the left and right sides.
• The task is to manipulate the equation while keeping this balance intact.

The addition property of equality allows us to add or subtract the same number from both sides. This ensures that the equation remains balanced, which is the essence of solving equations. By applying this property, the equation becomes simpler and easier to resolve in subsequent steps.

Isolate the Variable

Isolating the variable is an essential step in solving equations. It involves rearranging the equation so that the variable appears on its own on one side of the equation. In our example:

• The equation starts as \(6x - 5 = 19\).
• We need to first get rid of the \(-5\) next to \(6x\) by using the addition property of equality.
• Add 5 to both sides to cancel out the \(-5\), resulting in \(6x = 24\).

With this rearrangement, you can clearly see how close you are to finding \(x\). Isolating the variable involves precise operations to reduce the problem step by step, making it simpler until the variable stands alone.

Prealgebra Concepts

Prealgebra is the branch of mathematics that prepares students for algebraic thinking. It includes understanding basic equation-solving techniques, such as the addition property of equality. The concept of balancing equations is introduced here as:

• Applying operations equally on both sides ensures equations retain their equality.
• Breaking down complex equations into smaller, more manageable parts.

Prealgebra also introduces other fundamental concepts essential for further math studies, such as integers, fractions, and basic arithmetic operations. Mastering these concepts builds the foundation for tackling more advanced topics in algebra and beyond. By practicing these basic principles, you gain confidence and improve your ability to solve more complex mathematical problems.