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Chapter 6: Problem 94

Solve formula for the specified variable. \(-3 t-\frac{4}{p}=\frac{6}{s}\) for \(p\)

Short Answer

Expert verified
p = \frac{4}{-\frac{6}{s} - 3t}

Step by step solution

01

- Isolate the term containing the variable

Start by isolating the term that contains the variable we want to solve for, which is \( \frac{-4}{p} \). Add \( 3t \) to both sides of the equation to isolate the fraction: \[ -3t - \frac{4}{p} + 3t = \frac{6}{s} + 3t \]. This simplifies to \[ -\frac{4}{p} = \frac{6}{s} + 3t \]
02

- Eliminate the negative sign

Multiply both sides of the equation by \( -1 \) to get rid of the negative sign: \[ \frac{4}{p} = -\frac{6}{s} - 3t \].
03

- Cross-multiply to solve for p

To clear the fraction, multiply both sides of the equation by \( p \): \[ 4 = p(-\frac{6}{s} - 3t) \]
04

- Isolate p

Divide both sides by the expression \( -\frac{6}{s} - 3t \) to solve for \( p \): \[ p = \frac{4}{-\frac{6}{s} - 3t} \]

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Key Concepts

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

isolating variables
When you want to solve an equation for a specific variable, the first step is to isolate that variable on one side of the equation. This means moving all other terms to the opposite side of the equation. In the exercise given, we aimed to solve for the variable \( p \) in the equation \(-3 t-\frac{4}{p}=\frac{6}{s}\). To isolate \(-\frac{4}{p}\), we added \( 3t \) to both sides, resulting in the equation \(-\frac{4}{p} = \frac{6}{s} + 3t\). Isolating variables often involves basic arithmetic operations like addition, subtraction, multiplication, and division. These operations help to 'move' terms from one side of the equation to the other.
cross-multiplication
Cross-multiplication is a handy technique used to solve equations involving fractions. After isolating the fraction \(-\frac{4}{p}\) in the equation, we needed to get rid of the fraction to solve for \( p \). To achieve this, we multiplied both sides of the equation by \( p \), which gave us \( 4 = p \bigg(-\frac{6}{s} - 3t\bigg) \). This is known as cross-multiplication because we effectively 'cross' the denominator of one fraction to the numerator of the other side, simplifying the equation.
fractions in equations
Fractions can make equations look complex, but they're manageable with the right approach. Here are some tips:
  • To eliminate a fraction, you can often multiply both sides by the denominator of the fraction.
  • Be aware of negative signs and ensure you adjust them correctly by multiplying by \(-1\) if needed, as we did to simplify \(-\frac{4}{p}\) to \(\frac{4}{p}\).
In our exercise, after isolating the term with \( p \) and cross-multiplying, we solved for \( p \) by rearranging the equation to \( p = \frac{4}{-\frac{6}{s} - 3t} \). Solving equations with fractions often involves converting the equation into a form where the fraction is either eliminated or simplified.

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Most popular questions from this chapter

Multiply. Write each answer in lowest terms. $$ \frac{3 x}{x+3} \cdot \frac{(x+3)^{2}}{6 x^{2}} $$

Multiply or divide as indicated. Write each answer in lowest terms. $$ \frac{7}{12} \div \frac{15}{4} $$

Write each rational expression in lowest terms. $$ \frac{p+6}{p-6} $$

Rewrite each rational expression with the indicated denominator. $$ \frac{6}{k^{2}-4 k}=\frac{?}{k(k-4)(k+1)} $$

Multiply or divide. Write each answer in lowest terms. $$ \frac{(q-3)^{4}(q+2)}{q^{2}+3 q+2} \div \frac{q^{2}-6 q+9}{q^{2}+4 q+4} $$
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