Problem 18 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The solution to the equation is \( x = -8 \).

Step 1: Divide both sides of the equation by -7/2

In order to isolate x, the equation is divided by -7/2, as follows: \( x = \frac{28}{-\frac{7}{2}} \).

Step 2: Simplify the equation

The division by a fraction is the same as the multiplication by its reciprocal. We can simplify this calculation by multiplying 28 by -2/7, which gives: \( x = 28 \times -\frac{2}{7} \)

Step 3: Calculate the result

Multiply 28 by -2/7 to get the value of x: \( x = -8 \)

Step 4: Check the proposed solution

Substitute -8 for x in the original equation to verify that both sides of the equation are indeed equal: \( 28 = -\frac{7}{2} \times -8 \)

Key Concepts

Multiplication Property of Equality

When solving algebraic equations, it's essential to understand that what you do to one side of the equation, you must also do to the other. The multiplication property of equality states that if you multiply both sides of an equation by the same nonzero number, the two sides remain equal. This principle allows us to scale the equation up or down without changing its solution.

For example, if we have the equation \( 28 = -\frac{7}{2} x \) and we want to isolate the variable \( x \), we can multiply both sides of the equation by the reciprocal of \( -\frac{7}{2} \) (which we will discuss later in the Reciprocal of a Fraction section) without affecting the equality. This step is what prepares us for the simplification process where the variable \( x \) can be more easily isolated and solved for.

Isolating Variables

Isolating the variable, commonly the primary objective in solving an equation, means manipulating the equation in such a way that you end up with the variable on one side and numbers on the other. The goal is to have the variable alone, clearly showing its solution. In the given exercise, to isolate \( x \), we divide both sides of the equation by \( -\frac{7}{2} \) which is the coefficient of \( x \).

By applying the multiplication property of equality and using the reciprocal of the fraction, we are essentially performing the opposite mathematical operation to remove the coefficient from the variable. This reverse operation is what nudges us towards the solution as the variable \( x \) appears isolated after the simplification step.

Simplifying Equations

Simplifying an equation makes it easier to solve by reducing it down to its most basic form. One way to simplify an equation is by eliminating fractions, which we often do by multiplying by the reciprocal of the coefficient of the variable. To simplify \( x = \frac{28}{-\frac{7}{2}} \), we multiply by the reciprocal of \( -\frac{7}{2} \) which is \( -\frac{2}{7} \), thereby turning the division by a fraction into a multiplication by an integer. This ultimately gives us \( x = 28 \times -\frac{2}{7} \).

The simplification allows us to perform the operation more straightforwardly since multiplying by a whole number is simpler than dividing by a fraction. The simplicity gets us one step closer to uncovering the value of the variable.

Reciprocal of a Fraction

The reciprocal of a fraction is created by flipping the numerator and denominator. For instance, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \), assuming neither \( a \) nor \( b \) is zero as division by zero is undefined. When we divide by a fraction, it's equivalent to multiplying by its reciprocal, which is a fundamental concept in algebra. For our exercise, the reciprocal of \( -\frac{7}{2} \) is \( -\frac{2}{7} \).

We use this to simplify the equation as mentioned earlier. Applying the reciprocal is a valuable tool in algebra because it transforms division into multiplication, often making the arithmetic more manageable and leading to faster simplification in solving equations.