Problem 12 - Elementary Algebra

Question

In the following problems, solve each of the conditional equations. $$ 8 x=56 $$

Step-by-Step Solution

Answer

Answer: The value of x in the equation is 7.

Step 1: Write down the given equation

The given equation is: $$ 8x = 56 $$

Step 2: Isolate the variable x

To isolate x, we need to divide both sides of the equation by 8: $$ \frac{8x}{8} = \frac{56}{8} $$

Step 3: Simplify the equation

After dividing, we now have: $$ x = 7 $$ So, the solution to the equation is x = 7.

Key Concepts

Isolate the Variable

The process of solving an equation often involves the crucial step of isolating the variable. This means you need to manipulate the equation so that the variable you're solving for is on one side of the equation by itself. In our textbook example with the equation 8x = 56, the goal is to determine the value of x that makes the equation true.

To isolate x, we use the operation that will undo the multiplication by 8, which in this case is division. We divide both sides of the equation by 8. Using the division property of equality, which states that dividing both sides of an equation by the same non-zero number will not change the equality, we end up with the following:

\[\frac{8x}{8} = \frac{56}{8}\]

After performing the division, x is isolated on one side of the equation, thus making it clearer what x equals.

Equation Simplification

Simplifying an equation means reducing it to its most basic form while maintaining its equality. Simplification can involve various operations such as adding, subtracting, multiplying, dividing, and canceling like terms. In our sample problem, after isolating the variable x, we are left with a simpler form of the original equation:

\[x = \frac{56}{8}\]

This expression can still be simplified by performing the division. Simplification here is straightforward:

\[x = 7\]

Through simplification, the equation becomes clear and easily understandable. It is an important step since it makes verifying the solution elementary by substituting the found value back into the original equation.

Division Property of Equality

The division property of equality is a foundational principle in algebra that is used to solve equations. It asserts that if you divide both sides of an equation by the same number (except for zero), the balance of the equation remains unchanged, thus preserving the equality. In the example with the equation

\[8x = 56\]

to isolate x, we divide both sides by 8:

\[\frac{8x}{8} = \frac{56}{8}\]

By doing this, the left side simplifies to x, and the right side becomes 7:

\[x = 7\]

The division property of equality is crucial for solving equations because it allows us to manipulate and reduce equations in a way that leads to finding the solution.