Problem 12 - Algebra 1: Concepts and Skills

Question

Solve the equation by cross multiplying. Check your solutions. \(\frac{x}{5}=\frac{7}{3}\)

Step-by-Step Solution

Answer

The solution for the equation \(\frac{x}{5} = \frac{7}{3}\) is \(x = \frac{35}{3}\).

Step 1: Cross Multiplication

Perform cross multiplication which involves multiplying the numerator of one fraction with the denominator of the other. So, \(x * 3\) on the left side and \(5 * 7\) on the right side.

Step 2: Simplify the Equation

After cross multiplication, you get an equation \(3x=35\).

Step 3: Solve for x

Divide each side of the equation by 3 to isolate x. This gives the solution \(x = \frac{35}{3}\).

Step 4: Check the solution

Substitute \(x = \frac{35}{3}\) back into the original equation \(\frac{x}{5} = \frac{7}{3}\) to check if the equation holds true.

Key Concepts

Cross Multiplication Method

When facing equations with fractions, the cross multiplication method is a powerful tool to simplify the process. The method relies on the principle that when we have two fractions equal to each other, the cross products—or the product of the numerator of one fraction and the denominator of the other—are also equal. This is because the proportions are equivalent.
To apply this method, you simply multiply the top number (numerator) of one fraction with the bottom number (denominator) of the other fraction and do the same with the remaining numerator and denominator. Then, set these two products equal to each other to form a new equation without fractions, making it much easier to handle.
The beauty of cross multiplication is its simplicity and versatility; it can be applied to problems involving proportions, ratios, and direct comparisons of fractions. This technique streamlines solving for the unknown variable and is a cornerstone skill in algebra.

Algebraic Fractions

Algebraic fractions are simply fractions that include variables in their numerators, denominators, or both. Just like numerical fractions, algebraic fractions can represent parts of a whole or a ratio between two quantities.
Understanding how to work with algebraic fractions is vital, as they frequently appear in various mathematical problems and equations. The goals when handling these fractions include simplifying them to their lowest terms, performing arithmetic operations such as addition, subtraction, multiplication, and division, and solving for unknown variables.
Mastering algebraic fractions requires familiarity with basic fraction operations and skills such as factorization, common denominators, and simplifying expressions. Grasping these concepts not only boosts proficiency in algebra but also lays the groundwork for more advanced mathematics.

Isolating Variables

Isolating the variable is a fundamental technique used to solve algebraic equations. It involves manipulating the equation so that the variable we are solving for stands alone on one side of the equal sign. The other side of the equation should contain only numbers or simplified expressions.
To isolate a variable, you may need to use a variety of algebraic operations such as addition, subtraction, multiplication, division, and the distributive property. The objective is to reverse the operations surrounding the variable. For instance, if the variable is divided by a number, we multiply by that number to cancel out the division, and vice versa.
The goal of this process is to achieve an equation where the variable is clearly defined and equal to a numerical value or a simplified algebraic expression. Knowing how to expertly isolate variables is key to solving more complex equations efficiently.

Checking Algebraic Solutions

After solving an algebraic equation and finding the value of the variable, it is crucial to validate that the solution is correct. This verification process involves substituting the found value back into the original equation to see if it produces a true statement.
When you check your solution, if both sides of the original equation balance out—meaning the left-hand side equals the right-hand side—you have the correct solution. If they do not balance, it indicates a mistake somewhere in the solution process, and you'll need to retrace your steps to find the error.
Checking solutions not only confirms the correctness of your answers but also helps you understand the logic behind the mathematical operations you used. It's a habit that fosters accuracy and critical thinking in problem-solving.