Problem 71 - Algebra 1: Concepts and Skills

Question

Solve the equation using the cross product property. Check your solutions. $$ \frac{6}{x}=\frac{7}{x-5} $$

Step-by-Step Solution

Answer

The solution to the equation \(\frac{6}{x}=\frac{7}{x-5}\) is \(x=-30\).

Step 1: Cross Multiply

To make calculation easier, use the cross product property (the principle of cross multiplication) which states that the product of the extremes equals the product of the means for any proportion. Apply this to the given equation: \(6*(x-5) = 7*x\).

Step 2: Simplify

Simplify both sides of the equation. This results in: \(6x - 30 = 7x\).

Step 3: Solve for 'x'

Rearrange terms to solve for 'x'. Substract \(6x\) from both sides: \(-30 = 7x - 6x\), which simplifies to: \(-30 = x\). Negative x-values are often unexpected, so pay attention in this step.

Step 4: Check the solution

In order to affirm that the solution is correct, substitute \(-30\) for x in the original equation. We get \(\frac{6}{-30}=\frac{7}{-30-5}\) which simplifies to \(-0.2= -0.2\). As we see, both sides are equal, therefore, the solution is correct.

Key Concepts

Cross Multiplication

Cross multiplication is a powerful method used to solve rational equations, especially when you're dealing with a proportion. A proportion is an equation that states two ratios are equivalent. For example, in the equation \(\frac{a}{b} = \frac{c}{d}\), we're saying that the fraction \(\frac{a}{b}\) is equal to \(\frac{c}{d}\). To solve this using cross multiplication, we multiply the numerator of each fraction by the denominator of the other fraction. This results in the equation \(a \cdot d = b \cdot c\). By doing this, we eliminate the fractions and simplify the problem to a simple algebraic equation.In our exercise, cross multiplication was applied to \(\frac{6}{x} = \frac{7}{x-5}\), which resulted in:

• \(6 \cdot (x-5) = 7 \cdot x\)

This step transforms our fractional equation into a more manageable form.

Proportion

Proportion is a fundamental concept in mathematics that describes the equality of two ratios. When two ratios are set equal, like \(\frac{6}{x} = \frac{7}{x-5}\), you have a proportion. This indicates that there is a consistent relationship between the quantities involved, in this case, between 6 and 7, and their respective counterparts.Proportions are widely used because they solve for unknowns in situations where the relationship between numbers is consistent. In this context, once you establish that two fractions are equal, it opens the door to solving the equation through methods like cross multiplication.Cross multiplication utilizes the concept of proportion by asserting that the product of the means equals the product of the extremes, allowing us to solve for the unknown variable.

Algebraic Equations

Algebraic equations are expressions that contain numbers, variables, and operators like addition, subtraction, multiplication, and division. When we're solving these equations, we're essentially finding the value of the variable that makes the equation true.In our exercise, after applying cross multiplication, we simplified the equation to \(6x - 30 = 7x\). This is a linear algebraic equation because it involves powers of the variable \(x\) that are just 1.To solve it, you perform operations like combining like terms and isolating the variable on one side of the equation. For instance, the solution involves subtracting \(6x\) from both sides:

• \(-30 = 7x - 6x\)
• This simplifies to: \(-30 = x\)

Finally, checking the solution by substituting \(x\) back into the original equation ensures that both sides match, validating that our solution makes the equation true.