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Chapter 6: Problem 7

Solve each equation for \(y\). $$h=\frac{b}{y}+3$$

Short Answer

Expert verified
y = \frac{b}{h - 3}

Step by step solution

01

- Isolate the term containing y

First, subtract 3 from both sides of the equation to isolate the term with y on one side.\[ h - 3 = \frac{b}{y} \]
02

- Eliminate the fraction

Next, multiply both sides of the equation by y to get rid of the fraction.\[ y(h - 3) = b \]
03

- Solve for y

Finally, divide both sides of the equation by (h - 3) to isolate y.\[ y = \frac{b}{h - 3} \]

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Key Concepts

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

Isolate Variables
To solve an equation for a specific variable, one essential step is to isolate that variable on one side of the equation. This means you want the variable you are solving for (in this case, y) to be alone on one side of the equation.
This often involves several mathematical operations such as addition, subtraction, multiplication, or division.
Initially, you'll need to move other terms to the opposite side of the equation using inverse operations. For example, if the equation is: \(h = \frac{b}{y} + 3\)
First, you subtract 3 from both sides:
\(h - 3 = \frac{b}{y}\).
Now, you have successfully isolated the term with the variable (y).
Linear Equations
Understanding linear equations is critical in solving for a variable. A linear equation is one in which the highest power of any variable is one.
This means there are no exponents other than 1 and no higher-degree terms.
The general form of a linear equation in one variable is: \(ax + b = 0\), where a and b are constants, and x is the variable.
In our exercise, the term involving \(y\) can be manipulated to fit this form.
Once we move to:\(h - 3 = \frac{b}{y}\), we recognize that it's rearranged to isolate 'y'.
Linear equations can often be solved in straightforward steps, and understanding their structure helps simplify complex problems.
Fraction Elimination
Eliminating fractions simplifies solving equations. Fractions can make equations look more complicated than they are.
To eliminate a fraction, you perform operations that clear the denominator.
In our step-by-step solution, after isolating the term with y: \(h - 3 = \frac{b}{y}\), we multiply both sides by y:\(y(h - 3) = b\).
This operation removes the fraction, making it easier to solve for y.
To finalize, dividing both sides by \((h - 3)\) isolates \(y\):
\(y = \frac{b}{h - 3}\).
Understanding how to handle and eliminate fractions is essential for solving various types of equations easily and correctly.

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Most popular questions from this chapter

Solve each equation. $$\frac{y}{3}=\frac{27}{y}$$

Perform the indicated operations. Reduce answers to their lowest terms. See Example \(I\) $$ \frac{x^{2}+4 x-6}{x^{2}-9}-\frac{x^{2}+2 x-12}{x^{2}-9} $$

Perform the indicated operations. When possible write down only the answer. $$(a-b) \div(-1)$$

Solve each problem. It takes Gina 90 minutes to file the monthly invoices. If Hilda files twice as fast as Gina does, how long will it take them working together?

Find the domain of each rational expression. $$\frac{2 z-5}{7 z}$$
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