Problem 8 - Elementary Algebra

Question

Solve each of the equations. $$\frac{x-6}{7}=\frac{x+9}{8}$$

Step-by-Step Solution

Answer

\(x = 111\)

Step 1: Cross Multiply the Fractions

To eliminate the fractions, cross multiply. This means multiplying the numerator of the first fraction by the denominator of the second fraction and vice versa. You will have the equation:\[(x-6) \cdot 8 = (x+9) \cdot 7\]

Step 2: Expand Both Sides

Distribute the multiplication over the subtraction and addition within the parentheses:\[8(x-6) = 7(x+9)\]Expanding both sides gives:\[8x - 48 = 7x + 63\]

Step 3: Isolate the Variable

To solve for \(x\), first get all the \(x\) terms on one side of the equation. Subtract \(7x\) from both sides:\[8x - 7x - 48 = 63\]This simplifies to:\[x - 48 = 63\]

Step 4: Solve for the Variable

Now, add 48 to both sides of the equation to isolate \(x\):\[x - 48 + 48 = 63 + 48\]This simplifies to:\[x = 111\]

Key Concepts

Cross Multiplication

Cross multiplication is a technique used to solve equations involving fractions. It enables you to eliminate the fractions by rearranging the equation, making it easier to solve.

When you encounter an equation with a fraction on each side, you can use cross multiplication to transform it into a simpler expression. The process involves multiplying the numerator of one fraction by the denominator of the other fraction, and vice versa. Here's the basic technique:

• Take the numerator of the first fraction and multiply it by the denominator of the second fraction.
• Then, take the numerator of the second fraction and multiply it by the denominator of the first fraction.

This gives you a new equation without fractions, which is usually much easier to solve. For example, in the equation \(\frac{x-6}{7} = \frac{x+9}{8}\), cross multiplying yields \((x-6) \cdot 8 = (x+9) \cdot 7\). This is a crucial step in tackling algebra problems involving fractions.

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra. Once you have an equation without fractions, like in the previous example, solving it involves a few key steps.

First, simplify both sides of the equation as much as possible. This often involves using arithmetic operations like addition, subtraction, multiplication, or division. As seen above:

• Combine like terms. For instance, bringing all terms involving \(x\) to one side helps simplify your calculations.
• Perform the same operation to both sides of the equation to maintain equality.

Doing this helps to isolate terms involving \(x\), simplifying your path to finding the solution.

Next, you isolate the variable you're solving for by moving all other terms to the opposite side, as shown in the transformation from \(8x - 7x - 48 = 63\) to \(x - 48 = 63\). The ultimate goal is to have \(x\) (or the variable of interest) on one side, with a numerical value on the other, eventually solving it for \(x = 111\). Understanding these steps is critical to working with more complex algebraic equations.

Distributive Property

The distributive property is one of the essential properties of algebra. It's especially useful when dealing with expressions that involve parentheses.

This property states that multiplying a sum or difference by a number is the same as multiplying each addend by the number and then adding or subtracting the products. This principle is expressed as:

• \(a(b + c) = ab + ac\)
• \(a(b - c) = ab - ac\)

In our equation \(8(x-6) = 7(x+9)\), the distributive property helps to expand both sides:

• Distribute \(8\) over \((x-6)\) to get \(8x - 48\).
• Distribute \(7\) over \((x+9)\) to obtain \(7x + 63\).

By applying the distributive property, you ensure that each term inside the parentheses is counted in calculations. This maintains the equation balance and helps simplify terms in preparation for solving.