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Chapter 2: Problem 20

Solve each equation for \(y.\) $$ 8 x-3 y=4 $$

Short Answer

Expert verified
y = -4/3 + 8/3 x

Step by step solution

01

Move the term with y to the other side

Start by isolating the term with the variable y. Subtract 8x from both sides of the equation: 8x - 3y - 8x = 4 - 8x This simplifies to: -3y = 4 - 8x
02

Solve for y by dividing

Divide both sides of the equation by -3 to isolate y. Yy -3)y/-3 = (4 - 8x)/-3
03

Simplify the equation

Simplify the right-hand side of the equation to get y on its own. -3y 梅 -3 = (4 - 8x) 梅 -3Y = -4/3 + 8/3x

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Key Concepts

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

Isolation of Variables
Understanding how to isolate variables is crucial in solving linear equations. The goal is to get the variable you are solving for, in this case, \(y\), by itself on one side of the equation.

Let's revise the example from the exercise step by step:

Initial equation: 8x - 3y = 4

To isolate \(y\), we need to get rid of the term \(8x\) on the left side. The first step is to subtract \(8x\) from both sides of the equation: 8x - 3y - 8x = 4 - 8x

This simplifies to: -3y = 4 - 8x

Thus, we have successfully moved the term containing \(y\) to one side, isolating it for the next steps.
Simplification
Once variables are isolated, the next step is to simplify the equation as much as possible to solve for the variable. Simplification in linear equations typically involves performing basic arithmetic operations (addition, subtraction, multiplication, or division).

Continuing from our isolated equation: -3y = 4 - 8x

To solve for \(y\), we need to divide every term by \(-3\): -3y / -3 = (4 - 8x) / -3

This simplifies to: y = -4/3 + 8/3x

The division might seem tricky initially, but breaking it down into smaller steps can help. First divide both sides of the equal sign, and carefully handle negative signs.
Linear Equation Manipulation
Linear equation manipulation involves using algebraic operations to transform and simplify equations. Here are some key techniques:
  • Addition/Subtraction: Adding or subtracting the same value from both sides of an equation.
  • Multiplication/Division: Multiplying or dividing both sides of an equation by the same non-zero value.


In our example: 8x - 3y = 4

We started by subtracting \(8x\) from both sides to isolate \(y\): -3y = 4 - 8x

Then we divided by \(-3\) to solve for \(y\): y = -4/3 + 8/3x

These steps involve manipulating the equation to maintain equality while isolating and solving for the desired variable. Once you understand these techniques, solving linear equations becomes straightforward and manageable.

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Most popular questions from this chapter

Use the variable x for the unknown, and write an equation representing the verbal sentence. Then solve the problem. See Example 1. The sum of a number and \(-4\) is \(18 .\) Find the number.

Work Exercises 41鈥�45 in order. Solve the linear equation \(5(x+3)-2(x-4)=2(x+7),\) and graph the solution set on a number line.

When a consumer loan is paid off ahead of schedule, the finance charge is less than if the loan were paid off over its scheduled life. By one method, called the rule of \(78,\) the amount of unearned interest (the finance charge that need not be paid) is given by $$ u=f \cdot \frac{k(k+1)}{n(n+1)} $$ In the formula, u is the amount of unearned interest (money saved) when a loan scheduled to run for n payments is paid off k payments ahead of schedule. The total scheduled finance charge is \(f .\) Use the formula for the rule of 78 to work Exercises \(37-40\) Sondra Braeseker bought a new car and agreed to pay it off in 36 monthly payments. The total finance charge was \(\$ 700 .\) Find the unearned interest if she paid the loan off 4 payments ahead of schedule.

For each compound inequality, decide whether intersection or union should be used. Then give the solution set in both interval and graph form. See Examples \(2-4\) and \(6-8 .\) $$ -3 x \leq-6 \text { or }-3 x \geq 0 $$

Solve each equation, and check your solution. $$ 0.006(x+2)=0.007 x+0.009 $$
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