Problem 25 - Prealgebra and Introductory Algebra

Question

Solve. $$\frac{5}{x+3}=\frac{3}{x-1}$$

Step-by-Step Solution

Answer

The solution of the equation is x=7.

Step 1: Cross-Multiply

Cross-multiply the two fractions to get rid of the denominator. Follow the rule (a/b = c/d) => (a*d = b*c). This gives \(5 \cdot (x-1) = 3 \cdot (x+3)\).

Step 2: Simplify the Equation

Expand and simplify the equation by applying the distributive law. This gives \(5x - 5 = 3x + 9\).

Step 3: Solve for x

Get all terms involving x on one side and constant terms on the other side by subtracting 3x from both sides and adding 5 on both sides. So, it will give \(5x - 3x = 9 + 5\), which simplifies to \( x = 7\).

Key Concepts

Cross-Multiplication

Cross-multiplication is a widely used technique in algebra for solving equations that involve fractions, also known as algebraic fractions. It's particularly helpful when you are trying to find the value of a variable within an equation set up as a proportion.

Let's say you encounter an equation \(\frac{a}{b} = \frac{c}{d}\). To solve for the variable, one way to proceed is cross-multiplication, where you multiply the numerator of one fraction by the denominator of the other fraction and vice versa, setting both products equal to each other. If we apply this to our equation, we get \(a \cdot d = b \cdot c\).

For example, in the exercise \(\frac{5}{x+3} = \frac{3}{x-1}\), by cross-multiplying we get \(5 \cdot (x-1) = 3 \cdot (x+3)\), effectively removing the fractions and creating an easier equation to solve. Cross-multiplication must be done carefully to avoid loss of any terms or errors in signs.

Distributive Law

The distributive law, also known as distributive property, is an essential algebraic rule that comes in handy when you have to expand an expression. This law states that when a number is multiplied by a sum or difference, you can 'distribute' the multiplication to each term within the parentheses.

For our equation, \(5 \cdot (x-1)\) and \(3 \cdot (x+3)\) are both expressions where the distributive law can be applied. The law tells us to multiply every term inside the parentheses by the number outside, so we expand the expressions: \(5x - 5\) and \(3x + 9\), respectively.

Why Use Distributive Law?

The distributive law makes equations simpler and easier to manage, especially when solving for a variable. It's especially useful in the next steps, where you'll be gathering like terms and isolating variables to solve the equation.

Algebraic Fractions

Algebraic fractions are simply fractions where the numerator, the denominator, or both, contain algebraic expressions. These expressions can involve variables, constants, and exponents. In equations, solving algebraic fractions often involves finding a common denominator or using methods like cross-multiplication to eliminate the fractions altogether.

Typically, when you solve an equation with algebraic fractions, you want to get rid of the fractions early in the process to simplify your calculations. In our example, \(\frac{5}{x+3} = \frac{3}{x-1}\), each side of the equation is an algebraic fraction. By applying cross-multiplication, we turned it into a linear equation without fractions.

Working with algebraic fractions requires a good understanding of basic algebra operations, like addition, subtraction, multiplication, division, and the ability to manipulate and simplify algebraic expressions. These skills enable students to tackle a variety of algebra problems with confidence.