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Chapter 2: Problem 72

Solve each equation for \(y\). $$ 5 x-2 y-1=0 $$

Short Answer

Expert verified
\(y = \frac{5}{2}x - \frac{1}{2}\)

Step by step solution

01

Isolate Terms with y

Start by rearranging the equation to get all terms containing the variable \( y \) on one side. In this case, add \( 1 \) to both sides of the equation to get:\[5x - 2y = 1\]
02

Move Other Terms

Subtract \( 5x \) from both sides to isolate terms with \( y \) on the left side. This will give:\[-2y = -5x + 1\]
03

Solve for y

Divide every term in the equation by \(-2\) to solve for \( y \). This will simplify to:\[y = \frac{5}{2}x - \frac{1}{2}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Equations
Solving equations is a fundamental skill in algebra. It involves finding the value of the variable that makes the equation true. Here, we're focusing on solving for the variable \( y \), which means we want \( y \) on one side of the equation by itself. Once isolated, the equation is solved for \( y \). To solve an equation effectively:
  • Recognize that you aim to isolate the variable (in this case, \( y \)).
  • Use inverse operations, such as addition/subtraction and multiplication/division, to "undo" each operation on \( y \). This ensures you gradually strip away everything but \( y \).
  • Maintain balance by ensuring whatever you do to one side of the equation, you do to the other.
Understanding solving equations helps in many areas of mathematics, as it is the basis for finding unknowns.
Algebraic Manipulation
Algebraic manipulation refers to rearranging and simplifying expressions and equations using various techniques. This often involves operations like adding, subtracting, multiplying, or dividing both sides of an equation to reveal insights or simplify solutions.
In our exercise, manipulating the terms was crucial. By moving subtracted and multiplied terms across the equation, we simplify and align everything to view the solution more clearly. Here's how:
  • Identify terms that can be moved to isolate the target variable.
  • Use addition or subtraction to move constant terms across the equation.
  • Apply multiplication or division to simplify coefficients of terms associated with the variable.
These steps were essential in converting \(5x - 2y - 1 = 0\) into a more straightforward expression, eventually leading to \( y \) isolated.
Variables
Variables are symbols used to represent unknown values in mathematical equations and expressions. Commonly denoted by letters like \( x \) or \( y \), these variables can symbolize any number or value until solved.
In our example, both \( x \) and \( y \) are variables where we solve explicitly for \( y \). Understanding how to handle variables is essential:
  • Recognize that variables are placeholders in equations for values not yet known.
  • Treat operations with variables as you would with numbers: apply arithmetic operations while keeping equations balanced.
  • Consider the relationship between multiple variables; one variable might depend on another, as seen in our equation, \( y = \frac{5}{2}x - \frac{1}{2} \).
Mastering variables builds a strong foundation for more advanced algebraic concepts and problem-solving scenarios.

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Most popular questions from this chapter

Determine whether \((0,0)\) satisfies each inequality. Write \(y>|x|\)

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Graph each inequality. $$ y \geq 1 $$

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