Problem 63 - Elementary Algebra

Question

For Problems 55-70, solve each equation for the indicated variable. (Objective 4) $$ \frac{x-2}{4}=\frac{y-3}{6} \quad \text { for } y $$

Step-by-Step Solution

Answer

The solution for \( y \) is \( y = \frac{3x}{2} \).

Step 1: Cross-Multiply to Eliminate Fractions

To eliminate the fractions in the equation \( \frac{x-2}{4} = \frac{y-3}{6} \), we cross-multiply. This leads to the equation \( 6(x-2) = 4(y-3) \).

Step 2: Distribute on Both Sides

Distribute the 6 on the left side and the 4 on the right side of the equation: \( 6x - 12 = 4y - 12 \).

Step 3: Move Terms to Isolate y-Terms

Add 12 to both sides of the equation to simplify: \( 6x - 12 + 12 = 4y - 12 + 12 \), which simplifies to \( 6x = 4y \).

Step 4: Solve for y

To solve for \( y \), divide both sides of the equation by 4: \( y = \frac{6x}{4} \).

Step 5: Simplify the Expression

Simplify \( \frac{6x}{4} \) by dividing the numerator and the denominator by 2, which results in \( y = \frac{3x}{2} \).

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding values for variables that make an equation true. Equations can be made up of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, or division.
When you solve an equation, you work to isolate the variable, essentially "solving" what that variable equals. Here's a general approach:

• First, simplify both sides of the equation as much as possible. This might involve distributing terms or combining like terms.
• You then aim to get all terms involving the variable on one side of the equation, and constant terms on the other.
• Manipulate the equation by adding, subtracting, multiplying, or dividing both sides until you isolate the variable.

In the original exercise, the objective was to solve for the variable \( y \) in the equation \( \frac{x-2}{4}=\frac{y-3}{6} \). By performing arithmetic operations systematically, you can find that \( y = \frac{3x}{2} \). This result shows what \( y \) needs to equal to satisfy the initial equation.

Variables

In algebra, variables are symbols that represent unknown or changeable values. Most often, variables are denoted by letters like \( x \), \( y \), \( a \), or \( b \).
They are placeholders for numbers in equations and expressions, which allows us to form general rules applicable to many different scenarios. Here are some important points about variables:

• Variables can represent a range of values, including integers, fractions, and irrational numbers.
• They enable the formulation of equations that model real-life problems.
• Understanding how to manipulate variables is key to mastering algebra.

In the exercise, the variables \( x \) and \( y \) were used in an equation that lets you explore the relationship between the two. By replacing these symbols with specific numbers, you'd be able to solve real-world problems within the same mathematical framework.

Cross-Multiplication

Cross-multiplication is a useful algebraic technique for solving equations that involve fractions. It helps to clear fractions from an equation, making it easier to solve.
Here's how cross-multiplication works:

• Given an equation of the form \( \frac{a}{b} = \frac{c}{d} \), you multiply across the equal sign, which means \( a \) is multiplied by \( d \), and \( b \) is multiplied by \( c \).
• The equation transforms to \( ad = bc \), eliminating the fractions.

This technique is powerful because it directly simplifies equations involving rational expressions. In the step-by-step solution, cross-multiplication efficiently dealt with fractions by transforming \( \frac{x-2}{4}=\frac{y-3}{6} \) into \( 6(x-2) = 4(y-3) \). This allowed the problem solver to focus on the simpler task of distributing terms and solving a linear equation, rather than dealing with fractions directly. This method is invaluable in both algebra and more advanced mathematics.