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Chapter 2: Problem 11

Find the solution set of each equation. $$3 x-6 y=0$$

Short Answer

Expert verified
The solution set is y = \frac{x}{2}.

Step by step solution

01

Isolate y in the Equation

We start with the given linear equation: \[ 3x - 6y = 0 \]. Our goal is to solve for \( y \) in terms of \( x \). First, let's move the term involving \( x \) to the other side by subtracting \( 3x \) from both sides:\[ -6y = -3x \].Now, we need to get \( y \) by itself, so we'll divide everything by \( -6 \):\[ y = \frac{-3x}{-6} \].

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Key Concepts

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

Solving for Variables
In any linear equation, identifying and solving explicitly for a variable is a fundamental skill. It involves isolating the variable you want by rearranging the equation's terms until that variable stands alone on one side. In the given example \[ 3x - 6y = 0 \],your task is to solve for a specific variable, often represented as \( y \).Here鈥檚 a quick guide to tackle this:
  • Identify the variable to be isolated. In this case, it is \( y \).
  • Shift all terms containing other variables to the opposite side of the equation using addition or subtraction.
  • If your variable is still not isolated, remove any coefficients by performing the opposite operation, such as dividing if the coefficient was a multiplication.
These steps simplify your equation and help you explicitly state one variable in terms of the others, which is crucial in understanding how changes in one affect the others.
Equation Manipulation
Equation Manipulation is a technique where we rearrange terms in an equation to simplify or solve it. To manipulate a linear equation effectively, keep the following key points in mind:Understanding Inverse Operations:
  • Recognizing that subtraction is the inverse of addition and division is the inverse of multiplication can make it easier to move terms.
  • In the example \[ 3x - 6y = 0 \], subtracting \( 3x \) from both sides results in \[ -6y = -3x \].This operation helps in isolating terms on one side of the equation.
Maintaining Equality:
  • Remember that whatever operation you perform on one side of the equation, you must also execute on the other side to keep the equation balanced.For example, dividing all terms by \(-6\) in \[ -6y = -3x \] ensures that the equation remains equal, giving us \[ y = \frac{-3x}{-6} \].
With a firm grasp of equation manipulation, you'll find solving equations becomes much more intuitive and manageable.
Solution Sets
Once you've successfully solved for a variable, understanding the solution set for an equation is the next step. The solution set comprises all values that satisfy the equation when substituted back into its original form.For the manipulated equation, \( y = \frac{x}{2} \):
  • This represents a straight line in coordinate geometry consisting of all points \( (x, y) \) such that each point lies precisely on the line depicted by the equation.
  • The variable \( x \) can take any real number value, and for each \( x \), the appropriate \( y \) is \( \frac{x}{2} \).
This particular equation demonstrates that the solution set is a collection of endless pairs of \( (x, y) \) coordinates, forming a linear relationship between \( x \) and \( y \). Recognizing how solution sets can vary based on different equations can help you understand broader patterns in algebraic relationships.

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Most popular questions from this chapter

Balance the chemical equation for each reaction. $$\mathrm{FeS}_{2}+\mathrm{O}_{2} \longrightarrow \mathrm{Fe}_{2} \mathrm{O}_{3}+\mathrm{SO}_{2}$$

Use elementary row operations to reduce the given matrix to ( a) row echelon form and ( \(b\) ) reduced row echelon form. \(\left[\begin{array}{rrrr}2 & -4 & -2 & 6 \\ 3 & 1 & 6 & 6\end{array}\right]\)

Find the line of intersection of the given planes. $$3 x+2 y+z=-1 \quad \text { and } \quad 2 x-y+4 z=5$$

Prove the following corollary to the Rank Theorem: Let \(A\) be an \(m \times n\) matrix with entries in \(\mathbb{Z}_{p}\). Any consistent system of linear equations with coefficient matrix \(A\) has exactly \(p^{n-\text { rank }(A)}\) solutions over \(\mathbb{Z}_{p}\).

Set up and solve an appropriate system of linear equations to answer the questions. The process of adding rational functions (ratios of polynomials by placing them over a common denominator is the analogue of adding rational numbers. The reverse process of taking a rational function apart by writing it as a sum of simpler rational functions is useful in several areas of mathematics; for example, it arises in calculus when we need to integrate a rational function and in discrete mat hematics when we use generating functions to solve recurrence relations. The decomposition of a rational function as a sum of partial fractions leads to a system of linear equations. Find the partial fraction decomposition of the given form. (The capitall) letters denote constants. $$\frac{x^{2}-3 x+3}{x^{3}+2 x^{2}+x}=\frac{A}{x}+\frac{B}{x+1}+\frac{C}{(x+1)^{2}}$$
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