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Chapter 11: Problem 25

For Exercises 21 to \(32,\) solve for \(y\). $$3 x+2 y=6$$

Short Answer

Expert verified
The solution for \(y\) in the equation is \(y = 3 - \frac{3}{2}x\).

Step by step solution

01

Subtract \(3x\) from both sides

Subtract \(3x\) from both sides of the equation to isolate \(2y\) on one side. This gives \(2y = 6 - 3x\).
02

Divide by 2

Next, divide every term in the equation by 2 to solve for \(y\). This gives \(y = 3 - \frac{3}{2}x\).

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Key Concepts

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

Algebraic Manipulation
Algebraic manipulation is a process of rearranging and simplifying equations to make them easier to solve or understand. When dealing with linear equations like \(3x + 2y = 6\), our goal is often to solve for one variable in terms of the other, which involves several manipulation steps. Here, we use operations such as addition, subtraction, multiplication, and division to both sides of the equation.

  • These operations help maintain the equality while simplifying the equation.
  • In the given problem, subtracting \(3x\) from both sides is an example of using subtraction as a tool for algebraic manipulation.
  • It's important to perform the same operation on both sides of the equation to keep it balanced.

Algebraic manipulation serves as the foundational skill needed to effectively handle and solve equations, which is crucial in many areas of mathematics.
Isolating Variables
Isolating variables is a critical concept in solving equations. This technique involves rearranging an equation so that one variable is alone on one side of the equation. In the context of the exercise, we aim to solve for \(y\), meaning we're isolating \(y\) by getting rid of other terms on its side of the equation.

  • Initially, we subtract \(3x\) from both sides of the equation \(3x + 2y = 6\), leaving us with \(2y = 6 - 3x\).
  • To further isolate \(y\), we divide every term by 2, simplifying the equation to \(y = 3 - \frac{3}{2}x\)

This step-by-step process allows us to express \(y\) solely in terms of \(x\). This makes solving the equation simpler and more straightforward. Understanding how to isolate variables efficiently is especially useful when working with more complex systems of equations.
Two-Variable Equations
Two-variable equations are equations that include two different variables, in this case, \(x\) and \(y\). The main goal when solving these equations is usually to express one variable in terms of the other. This allows us to explore the relationship between the two variables stringently.

  • A common method to solve such equations is to isolate a variable (like we did with \(y\)), facilitating easier manipulation and interpretation of the equation.
  • This can often provide insights into the nature of the relationship, such as determining a slope or a specific point of intersection when graphing.
  • Working with two-variable equations helps build understanding of linear relationships and prepares for more advanced subjects such as coordinate geometry and calculus.

Solving for one variable helps in visualizing these relationships. This is key for graphical interpretations or further calculations involving these variables.

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Most popular questions from this chapter

Simplify. $$\frac{4 x-3}{3 x^{2} y}+\frac{2 x+1}{4 x y^{2}}$$

Suppose that you drive about \(12,000 \mathrm{mi}\) per year and that the cost of gasoline averages 3.70 dollar per gallon. a. Let \(x\) represent the number of miles per gallon your car gets. Write a variable expression for the amount you spend on gasoline in one year. b. Write and simplify a variable expression for the amount of money you will save each year if you increase your gas mileage by 5 miles per gallon. c. If you currently get 25 miles per gallon and you increase your gas mileage by 5 miles per gallon, how much will you save in one year?

State whether the given division is equivalent to \(\frac{x^{2}-3 x-4}{x^{2}+5 x-6}\). $$\frac{x+1}{x+6} \div \frac{x-1}{x-4}$$

Divide. $$\frac{2 x^{2}-3 x-20}{2 x^{2}-7 x-30} \div \frac{2 x^{2}-5 x-12}{4 x^{2}+12 x+9}$$

State whether the given division is equivalent to \(\frac{x^{2}-3 x-4}{x^{2}+5 x-6}\). $$\frac{x+1}{x-1} \div \frac{x+6}{x-4}$$
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