Problem 32 - Elementary Algebra

Question

Solve the equations. $$ \frac{9 x}{7}=6 $$

Step-by-Step Solution

Answer

Answer: x = $\frac{42}{9}$ (approximately 4.67)

Step 1: Clear the fraction

To clear the fraction, we need to multiply both sides of the equation by 7, which is the denominator. This will give us: $$ 7 \cdot \frac{9 x}{7} = 7 \cdot 6 $$

Step 2: Simplify the equation

Now, we simplify the equation by canceling the 7 from the numerator and the denominator: $$ 9x = 42 $$

Step 3: Isolate variable x

Next, we want to isolate x by dividing both sides of the equation by the coefficient of x, which is 9: $$ \frac{9x}{9} = \frac{42}{9} $$

Step 4: Find the value of x

Simplifying the equation further, we get the value of x: $$ x = \frac{42}{9} $$ This is the final solution in fractional form. If you want to express the answer as a decimal, you can simply divide 42 by 9 to get approximately 4.67.

Key Concepts

Clearing Fractions

When you encounter a linear equation with fractions, one of the first steps is often 'clearing the fractions' to simplify the solving process. This technique involves getting rid of the fraction by multiplying both sides of the equation by the least common denominator (LCD).
For example, in the equation \( \frac{9x}{7} = 6 \), the number 7 is the only denominator, so it's also our LCD. By multiplying both sides by 7, the fraction is effectively cleared: \( 7 \cdot \frac{9x}{7} = 7 \cdot 6 \), which simplifies to \(9x = 42\). This leaves us with a much simpler equation to solve, without the added complexity of dealing with fractions.

Isolating Variables

After clearing fractions, the next step is usually to isolate the variable. This means rearranging the equation so that the variable we are solving for is on one side of the equation by itself.
In the given problem, after clearing the fraction, we are left with \(9x = 42\). To isolate \(x\), we divide both sides of the equation by 9: \( \frac{9x}{9} = \frac{42}{9} \). This gives us \(x = \frac{42}{9}\), with the variable \(x\) now isolated. Understanding how to manipulate equations to isolate variables is crucial for solving not only linear equations but many other types of equations in algebra.

Simplifying Equations

Simplifying equations is a fundamental process in algebra that involves reducing equations to their simplest form to make them easier to work with. This can include combining like terms, reducing fractions, and canceling out terms.
In the current problem, once we have isolated \(x\), the equation \( \frac{9x}{9} = \frac{42}{9} \) is already quite simple. However, we can simplify it further to find the fractional solution for \(x\). Reducing the fraction \(\frac{42}{9}\) gives us the simplest form of the solution. Simplifying equations not only makes them easier to solve but also makes it easier to understand the relationship between the variables involved.

Fractional Solutions

Sometimes, the solution to a linear equation is in the form of a fraction, which is perfectly valid. A fractional solution represents the exact value of the variable.
In our example, the final step of the solution process gave us \(x = \frac{42}{9}\). This fraction can be further simplified to \(x = \frac{14}{3}\), or left in decimal form as approximately 4.67. Whether you present your answer as a fraction or a decimal can depend on the context of the problem or instructions given. Fractions are often considered a more exact representation, while decimals may be used for an approximate or practical solution in real-world contexts.