Problem 26 - Prealgebra

Question

Using the addition property of equality first, solve each of the following equations. $$7 x-5=30$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 5\).

Step 1: Understand the Equation

The equation given is \(7x - 5 = 30\). Our goal is to solve for \(x\), meaning we want to rewrite the equation so that \(x\) is on one side by itself.

Step 2: Use the Addition Property of Equality

To eliminate the \(-5\) from the left side, we add \(5\) to both sides of the equation. This gives us: \[7x - 5 + 5 = 30 + 5\] Simplifying both sides, we have: \[7x = 35\].

Step 3: Solve for x by Using the Division Property of Equality

Next, we divide both sides of the equation by \(7\) to solve for \(x\): \[\frac{7x}{7} = \frac{35}{7}\] Simplifying, we obtain: \[x = 5\].

Key Concepts

Solving Linear Equations

Linear equations are fundamental in algebra, and they typically appear in the form of \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants, while \(x\) is the variable we need to solve for.
The primary goal in solving linear equations is to isolate the variable \(x\) on one side of the equation. This helps us determine its value by simplifying the equation step by step. Understanding the basic principles of how to manipulate equations is essential:

• We often start by eliminating any terms that are not connected to the variable. This can involve adding or subtracting numbers from both sides.
• Simplifying each step is critical to avoid mistakes and ensure clarity.

In the equation \(7x - 5 = 30\), our first move is to use the Addition Property of Equality to tackle the equation efficiently.

Equation Simplification

Equation simplification involves transforming an equation into a simpler or more convenient form. This is done through various mathematical operations that reduce the equation towards its solution, typically by performing operations like addition, subtraction, multiplication, and division in a strategic manner.
For the equation \(7x - 5 = 30\), simplification begins with adding \(5\) to both sides to cancel out the \(-5\) term adjacent to the variable \(x\):

• Add \(5\) to each side: \(7x - 5 + 5 = 30 + 5\)
• This simplifies to \(7x = 35\).

Simplifying the equation incrementally reinforces the step-by-step process of bringing it to a form where solving for the variable \(x\) becomes straightforward.

Division Property of Equality

The Division Property of Equality states that if you divide both sides of an equation by the same nonzero number, the two sides remain equal. This principle is particularly useful when solving for a variable like \(x\) that is currently multiplied by a number.
In our scenario of the equation \(7x = 35\), we apply this property to isolate \(x\). This is achieved by dividing all terms by the coefficient of \(x\), which in this case is \(7\):

• Divide both sides by \(7\): \(\frac{7x}{7} = \frac{35}{7}\)
• This gives \(x = 5\).

The division simplifies the equation directly to the solution, letting us find that the value of \(x\) is \(5\). Understanding how and when to apply the Division Property of Equality is crucial in solving not only linear equations but also more complex mathematical scenarios.