Problem 8 - Elementary Algebra

Question

For the following problems, solve the rational equations. $$ \frac{8}{y}=\frac{2}{3} $$

Step-by-Step Solution

Answer

Question: Solve the rational equation: \(\frac{8}{2y} = \frac{3}{3}\) for y. Answer: The solution to the given rational equation is \(y = 12\).

Step 1: Cross-multiply the fractions

Cross-multiplying the fractions in the given equation will eliminate the fractions so that we have a simpler linear equation. So, we multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa: \( (8 * 3) = (2 * y) \)

Step 2: Simplify the equation

Now, let's simplify both sides of the equation: \( 24 = 2y \)

Step 3: Solve for y

Since we have a linear equation, we can now solve for "y" by dividing both sides of the equation by 2: \( y = \frac{24}{2} \)

Step 4: Simplify and write the final answer

Finally, let's find the simplified value for "y": \( y = 12 \) The solution to the given rational equation is \( y = 12 \).

Key Concepts

Cross-Multiplication

Understanding cross-multiplication is vital when solving rational equations, which involve ratios or fractions. Cross-multiplication is a method used to solve an equation where two fractions are set equal to each other. By multiplying the numerator of one fraction by the denominator of the other fraction, and doing the same in reverse, the relationship between the fractions is maintained, yet it simplifies the equation greatly. This technique can remove the fractions entirely, leaving you with a simpler equation to solve.

To apply cross-multiplication, remember the following steps:

• Identify the numerators and denominators of the fractions in the equation.
• Multiply the numerator of the first fraction by the denominator of the second fraction.
• Do the same with the numerator of the second fraction and the denominator of the first fraction.
• Set these two products equal to each other to form a new equation without fractions.

Through practice, cross-multiplication becomes an intuitive step in managing equations with algebraic fractions.

Linear Equations

Once you've used cross-multiplication to clear out fractions, you will often find yourself dealing with a linear equation. Linear equations are algebraic equations in which each term is either a constant or the product of a constant and a single variable. Linear equations can typically be written in the form of \( ax + b = 0 \), where \( a \) and \( b \) are constants. They are called 'linear' because the graph of these equations is always a straight line.

The steps to solve a linear equation are:

• Simplify both sides of the equation if necessary by combining like terms and using the distributive property.
• Isolate the variable on one side of the equation by using addition, subtraction, multiplication, or division.
• Simplify further if needed.

Mastering linear equations is a fundamental algebra skill, as it sets the stage for solving more complex mathematical problems.

Algebraic Fractions

Algebraic fractions are simply fractions with algebraic expressions in the numerator, the denominator, or both. These can sometimes be intimidating due to their appearance, but they follow the same principles as arithmetic fractions.

To work with algebraic fractions, make sure to:

• Keep the equation balanced by performing any operation equally on both sides of the equation.
• Factor the expressions when possible to simplify the fractions.
• Find a common denominator to combine fractions or to solve equations involving multiple algebraic fractions.
• Be mindful of the domain of the function, as division by zero is undefined.

It's important to practice simplifying algebraic fractions and to understand operations involving them to navigate smoothly through algebraic problems.