Problem 4 - Algebra for College Students

Question

Solve each equation. \(\frac{-5 x}{4}=\frac{7}{2}\)

Step-by-Step Solution

Answer

The solution is \(x = \frac{-14}{5}\).

Step 1: Cross Multiply

To eliminate the fractions, cross multiply the terms: \((-5x) \times 2 = 7 \times 4\). This simplifies to \(-10x = 28\).

Step 2: Isolate x

Divide both sides of the equation by -10 to solve for \(x\). \(x = \frac{28}{-10}\).

Step 3: Simplify the Fraction

Simplify the fraction \(\frac{28}{-10}\) to its lowest terms. The greatest common divisor of 28 and 10 is 2, so divide both by 2: \(x = \frac{-14}{5}\).

Key Concepts

Cross Multiplication

Cross multiplication is a handy technique used to solve equations involving fractions. It helps to eliminate the fractions by multiplying across the equation. Here’s how it works in simple terms.First, identify the two fractions that need attention. In the original exercise, we have \[\frac{-5x}{4} = \frac{7}{2}\]To cross multiply these fractions, take the numerator of each fraction and multiply it by the denominator of the other fraction. Therefore, multiply \((-5x)\) by \(2\) and \(7\) by \(4\). This results in:\[(-5x) \times 2 = 7 \times 4\]Simplifying those products gives you a new equation without fractions:\[-10x = 28\]This eradicates the need to deal with separate fractions, making it easier to solve for the variable in the next steps.

Fraction Simplification

Simplifying fractions is an essential skill in mathematics, helping to reduce complexity in calculations and find the solution more easily. Let’s break this process down using the result from cross multiplication.After performing cross multiplication, you arrived at a fraction: \[x = \frac{28}{-10}\]To simplify a fraction, find the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both numbers without leaving a remainder.

• For 28 and 10, the GCD is 2.

Divide both the numerator and the denominator by this GCD:

• Numerator: \(28 \div 2 = 14\)
• Denominator: \(-10 \div 2 = -5\)

This results in a simplified fraction:\[x = \frac{-14}{5}\]This simplified form is easier to interpret and work with, ensuring you present your answers in the simplest way possible.

Isolation of Variable

The isolation of a variable is the process of solving for a specific variable within an equation. This technique makes an unknown variable stand alone on one side of the equation. In solving linear equations, it turns complex problems into simpler, manageable forms.In the original example, after obtaining the equation \[-10x = 28\], our goal is to solve for \(x\). To isolate \(x\), divide both sides of the equation by the coefficient attached to \(x\), which is \(-10\).This calculation looks like:\[x = \frac{28}{-10}\]By dividing, you effectively "cancel out" the \(-10\) from the left side, leaving \(x\) by itself:

• This operation is key for isolating \(x\) and reaching a solution.

After isolating \(x\), simplifying the result through fraction simplification provides the final solution. The isolated variable \(x\) is now clear and precisely valued, shown as:\[x = \frac{-14}{5}\]This organized approach makes the equation easier to handle and clear up any potential confusion.