Question

Solve the equation by cross multiplying. Check your solution(s). $$\frac{9}{3 x}=\frac{4}{x+2}$$

Short answer

The solution for the equation \(x = 6\) after cross-multiplication, simplification, and verification.

Step-by-step solution

Cross-Multiplication

Cross-multiplication is the process of canceling out denominators of fractions so that we can more easily solve the equation. In this case, cross-multiplication gives us \(9 \cdot (x+2) = 4 \cdot (3x)\).

Distribute and Simplify

Distribute the numbers on both sides to eliminate parenthesis and collect the like terms. Thus the equation becomes \(9x+18 = 12x\). Arrange the terms to get the equation into the form \(ax=b\), so the equation becomes \(3x = 18\).

Solve for the variable

Divide the equation by 3 on both sides to isolate the variable x. Therefore, \(x = 18/3\) implies \(x = 6\).

Check the solution

Substitute x=6 back into the original equation to verify that the left hand side equals the right hand side: \(\frac{9}{3*6}=\frac{4}{6+2}\), which simplifies to \(\frac{9}{18}=\frac{4}{8}\), or \(0.5=0.5\), showing this is valid.

Key concepts

Solving Equations

When we talk about solving equations, we're looking for a value that makes the equation true. In algebra, equations often involve variables, which are unknowns that we need to determine. Cross-multiplication is a useful method when dealing with equations that have fractions. To solve the given equation by cross-multiplication, \[ \frac{9}{3x} = \frac{4}{x+2}, \]we eliminate the fractions by multiplying the numerator of one side by the denominator of the other side and vice versa. This step gives us an equation without fractions:\[9 \cdot (x+2) = 4 \cdot (3x).\]After cross-multiplying, we distribute and combine like terms to simplify the equation further into a more solvable form, which can often be written simply as a linear equation like \(ax = b\). Once simplified, solving for the variable involves making the variable the subject of the equation. In this example, we found \(x = 6\) as the solution. Remember, it's essential to handle each algebraic step carefully to ensure that the solution remains valid.

Rational Expressions

Rational expressions are fractions that have polynomials in the numerator, the denominator, or both. In the equation we solved, we had rational expressions:\(\frac{9}{3x} \text{ and } \frac{4}{x+2}.\)These expressions can look complex, but by using cross-multiplication, we created a more manageable problem by simplifying it into a basic equation. This is crucial because it provides a straightforward way to work with otherwise complicated fractions. Here are steps you can follow to solve rational equations like these:

• Clear fractions by multiplying through by the least common denominator if necessary,
• Perform cross-multiplication to eliminate the denominators altogether,
• Simplify and solve the resulting equation.

Rational expressions require careful simplification and manipulation to avoid errors. Always plan to set denominators not equal to zero since division by zero is undefined. This aids in identifying any potential restrictions on the solution of the equation.

Checking Solutions

After solving an equation, it's always important to check your solutions. This step ensures that the solution makes the original equation true and that no mistakes were made during calculations.For our example, after finding that \(x = 6\), we plug it back into the original equation:\[ \frac{9}{3 \cdot 6} = \frac{4}{6+2}. \]When simplified, both sides give the value of 0.5, confirming our solution was correct.Checking solutions involves:

• Substituting the found value(s) back into the original equation,
• Comparing both sides of the equation to verify they are equal,
• Ensuring the solution does not violate any restrictions, like making any denominators zero.

This practice is a good habit to develop because it also helps catch calculation errors and understand whether the solution is feasible within the problem's constraints.