Problem 15 - Algebra 1

Question

Solve the equation by cross multiplying. $$\frac{x}{10}=\frac{14}{5}$$

Step-by-Step Solution

Answer

After calculating, the value of 'x' is 28.

Step 1: Cross Multiply

The concept of cross multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction and vice versa. In this case, harnessing this technique, \(x * 5=14 * 10\).

Step 2: Simplify the Equation

On simplification, the equation turns into \(5x = 140\). It is obtained by multiplying the numbers.

Step 3: Solve for x

Lastly, in order to find the value of 'x', divide both sides of the equation by 5. The equation becomes \(x = \frac{140}{5}\).

Key Concepts

Solving Equations

When we talk about solving equations, we refer to the process of finding the values of the variables that make the equation true. This process is central to algebra and is a skill used across all levels of mathematics. In our exercise, the equation is set up as a proportion with two fractions. By cross multiplying, we link the concept of fractions to solving equations. After cross multiplying, we obtain an equation in a more familiar format, specifically in the form of a simple linear equation, which we can solve by isolative operations. Isolative operations involve undoing the operations performed on the variable. In the example provided, the operation to be undone is multiplication by 5, so we perform the inverse operation--division by 5--to isolate and solve for the variable 'x'.

Fractions

The concept of fractions is essential when dealing with ratios and proportions in mathematics. Fractions consist of a numerator (top number) and a denominator (bottom number), representing a part of a whole. In our exercise, each side of the equation is a fraction. The relationship between these fractions is key to solving the equation. Understanding fractions is crucial here because it allows us to apply the technique of cross multiplication accurately and effectively. This technique manipulates the structure of fractions to create an easier equation to solve.

Simplifying Equations

Simplifying equations is a process that involves reducing equations to their simplest form to make them easier to solve. This usually involves combining like terms, canceling out terms where possible, and carrying out arithmetic operations. In the step by step solution we explored, once the initial cross multiplying is done, we simplify the resulting equation from a multiplication of two numbers to a single linear equation. Simplifying makes the equation look less complex and daunting, aiding us in solving for the variable 'x' with greater ease. It's important to do this step properly, as it sets the stage for the final operation to find the variable's value.