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Chapter 3: Problem 48

Solve for \(y\). a) \(4 y=36\) b) \(a y=x\) c) \(p y=r\)

Short Answer

Expert verified
a) \(y = 9\) b) \(y = \frac{x}{a}\) c) \(y = \frac{r}{p}\)

Step by step solution

01

Equation a: \(4 y=36\)

To solve for y, we will divide both sides of the equation by the coefficient of y, which in this case is 4: \(y = \frac{36}{4}\) Now, simplify the fraction: \(y = 9\) The solution for equation a is y = 9.
02

Equation b: \(a y=x\)

To solve for y, we will divide both sides of the equation by the coefficient of y, which in this case is a: \(y = \frac{x}{a}\) The solution for equation b is y = \(\frac{x}{a}\).
03

Equation c: \(p y=r\)

To solve for y, we will divide both sides of the equation by the coefficient of y, which in this case is p: \(y = \frac{r}{p}\) The solution for equation c is y = \(\frac{r}{p}\).

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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 an essential skill in solving linear equations. It's all about rearranging equations to make them easier to work with. This involves using properties of numbers and operations to change the equation without affecting its solutions.

Consider equation \(4y = 36\). When you're manipulating this equation, you're really just trying to express it in a simpler form. By dividing both sides by 4, you simplify the equation to \(y = 9\).
  • The goal is to maintain equality on both sides of the equation while isolating the variable you're solving for.
  • This process can include operations such as addition, subtraction, multiplication, and division.
Think of algebraic manipulation as a series of logical steps that allow you to peel away the layers of an equation to reach the solution.
Isolating Variables
Isolating variables is a crucial step when solving equations. It involves getting the variable you're solving for by itself on one side of the equation. This technique uncovers the value of the variable, giving you the answer you need.

In equations like \(ay = x\), our goal is to make \(y\) stand alone. This means taking the necessary operation, division in this case, to move \(a\) to the other side. So, divide both sides by \(a\) to isolate \(y\), giving \(y = \frac{x}{a}\).
  • Isolating variables often requires reversing operations that are applied to the variable.
  • The main aim is to have the variable appear by itself, often leading to easier interpretation of the solution.
Mastering this skill helps in tackling more complex equations where multiple manipulations might be needed.
Equation Solving Techniques
Equation solving techniques are strategies used to find the value of variables in equations. These techniques can vary, but they are all about finding a systematic approach to arrive at the solution.

Take the equation \(py = r\). By identifying the variable \(y\) that we need to solve, we can use division as our technique here. Dividing both sides by \(p\) reveals \(y = \frac{r}{p}\). This approach is often called solving by inspection, where you eyeball the operation needed to move coefficients or constants to the opposite side of the equation.
  • Equation solving can involve different techniques depending on the form of the equation.
  • Understanding the basics of arithmetic operations is crucial in applying these techniques effectively.
Developing a mental toolbox of these techniques allows you to approach a variety of mathematical problems with confidence.

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Most popular questions from this chapter

Solve using the five "Steps for Solving Applied Problems." See Examples \(7-9\) Three times a number is fifteen more than half the number. Find the number.

Solve each formula for the indicated variable. $$A=P+P R T\( for \)T$ (finance)$$

Objectives 2 and 3 Mixed Exercises Solve using the five "Steps for Solving Applied Problems." Henry marks up the prices of his fishing poles by \(50 \%\). Determine what Henry paid his supplier for his best-selling fishing pole if Henry charges his customers \(\$ 33.75\) (THE IMAGES CANNOT COPY)

Solve each equation by first clearing fractions or decimals. $$0.2(y-3)+0.05(y-10)=-0.1$$

Solve each formula for the indicated variable. $$I=P R T\( for \)P$ (finance)$$
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