Problem 19 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation using the multiplication property of equality. Be sure to check your proposed solutions. $$28=-\frac{7}{2} x$$

Step-by-Step Solution

Answer

The solution to the equation \(28=-\frac{7}{2} x\) is \(x=-8\).

Step 1: Identify the coefficient of the variable

In the equation \(28=-\frac{7}{2} x\), the coefficient of the variable x is -\(\frac{7}{2}\).

Step 2: Isolate the variable

To isolate the variable x, divide both sides of the equation by the coefficient of x, which is -\(\frac{7}{2}\). This gives: \(x = 28 / -\frac{7}{2}\).

Step 3: Simplify the Right Side

The division can be simplified to: \(x = 28 * -\frac{2}{7}\). Then, by simplifying, it gives: \(x = -8\)

Step 4: Check the solution

To check if -8 is the solution, substitute it back into the original equation: \(28 = - \frac{7}{2} * -8\). The left hand side equals the right hand side, so -8 is indeed a solution to the equation.

Key Concepts

Solving Equations

Understanding the process of solving equations is crucial. An equation states that two expressions are equal and our goal is to find the value of the variable that makes this statement true. In our example, we have the equation \(28 = -\frac{7}{2}x\). Here, \(x\) is the variable we need to solve for, and the expressions on both sides are \(28\) and \(-\frac{7}{2}x\). The challenge is to find out what \(x\) must be in order for these two expressions to be identical. When solving an equation, you often use various mathematical operations, ensuring that whatever you do to one side, you do to the other. This keeps the balance of the equation. An important property that often comes into play is the multiplication property of equality. This property tells us that you can multiply or divide both sides of an equation by the same non-zero number without changing the equation's solution. As we go through the problem-solving process, you'll see how this property helps us in finding \(x\).

Isolation of Variables

To solve an equation like \(28 = -\frac{7}{2}x\), you need to isolate the variable \(x\). Isolating a variable means getting \(x\) by itself on one side of the equation. This is typically the first major step in solving any algebraic equation.Start by identifying the coefficient of \(x\), which in this case is \(-\frac{7}{2}\). Our goal is to manipulate the equation so that \(x\) remains on one side, and everything else is on the other side. To achieve this, you can divide both sides of the equation by the coefficient of \(x\), \(-\frac{7}{2}\). When you divide \(28\) by \(-\frac{7}{2}\), you are actually multiplying \(28\) by the reciprocal of \(-\frac{7}{2}\), which simplifies to multiplying by \(-\frac{2}{7}\). This results in the expression: \[ x = 28 \times -\frac{2}{7} \]This manipulation using division and multiplication helps effectively move \(x\) to one side of the equation, free from its coefficient, allowing us to easily find its value.

Algebraic Manipulation

Algebraic manipulation involves using mathematical operations to rearrange and simplify equations or expressions in order to solve for the variable of interest.In our specific example, solving the equation \(28 = -\frac{7}{2}x\) requires strategic manipulation. After isolating \(x\) by dividing both sides by \(-\frac{7}{2}\) which is essentially multiplying by \(-\frac{2}{7}\), we simplify the expression to find the value of \(x\). This looks like:\[ x = 28 \times -\frac{2}{7} \]Next, perform the multiplication. Multiply \(28\) and \(-\frac{2}{7}\), resulting in:\[ x = -8 \]This demonstrates the importance of careful manipulation and simplification to ensure you arrive at the correct solution. More generally, algebraic manipulation is about understanding how to transform an equation effectively to focus on the variable. By practicing these steps, you boost your problem-solving skills and ability to work through a variety of algebraic equations.