Problem 16 - Prealgebra

Question

Use the multiplication property of equality to solve each of the following equations. In each case, show all the steps. $$7 x=-35$$

Step-by-Step Solution

Answer

x = -5

Step 1: Identify the equation

The given equation is: \(7x = -35\).

Step 2: Understand the multiplication property of equality

The multiplication property of equality states that if you divide both sides of an equation by the same non-zero number, the equation remains balanced and true. Here, we will use it to isolate \(x\) by dividing both sides by 7.

Step 3: Apply the property to solve for x

To isolate \(x\), divide both sides of the equation by 7: \(x = \frac{-35}{7}\).

Step 4: Simplify the equation

Divide \(-35\) by 7 to solve for \(x\), resulting in \(x = -5\). This is the simplified solution where \(x\) is isolated.

Key Concepts

Solving Equations

Solving equations can feel like solving a puzzle, where your goal is to find the value of the variable that makes the equation true. An equation is a mathematical statement that asserts equality between two expressions. In essence, solving an equation involves finding that unknown number which will satisfy this statement of equality. For example, consider the equation: \(7x = -35\). Here, \(x\) is the unknown variable we need to solve for. By finding the value of \(x\), you're balancing both sides of the equation, making them equal. The trick to solving equations is to perform operations that simplify one side of the equation without changing its equality. You can think of it as keeping your balance on a seesaw; whatever you do to one side, you do to the other. This ensures the equation remains valid and true.

Steps to Solve Equations

Solving equations involves a series of steps, much like following a recipe. Each step builds upon the previous one until you arrive at a solution. Let's look at these steps:

• First, **identify** the equation you are working with. In our example, it is \(7x = -35\).
• Next, **understand** the properties of equality you'll use. Here, we use the multiplication property of equality, which allows us to divide both sides of the equation by the same non-zero number to isolate the variable.
• Then, **apply** the property to the equation. We divide both sides by 7 in our example: \(x = \frac{-35}{7}\).
• Finally, **simplify** the equation to find the value of the variable. Dividing, we find \(x = -5\).

Following these steps helps you approach any equation systematically, reducing errors and ensuring a clear path to the solution.

Prealgebra Concepts

Diving into prealgebra, you encounter foundational concepts that set the stage for more complex math. These include understanding how numbers and variables interact in equations. A key prealgebra concept used in solving \(7x = -35\) is the multiplication property of equality, which is a fundamental principle in algebra. This property emphasizes that multiplying (or dividing) both sides of an equation by the same number does not affect the permanence of the equality. This property is crucial because it lays the groundwork for manipulating equations in a way that enables you to find the value of unknowns effectively. By mastering these basic properties, you build confidence, enabling you to progress to more advanced mathematical concepts with ease. Knowing how to apply these properties with confidence allows for efficient and effective problem-solving.