/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 39 Solve each equation. $$-\frac{... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 39

Solve each equation. $$-\frac{3}{7} x=4$$

Short Answer

Expert verified
x = -\frac{28}{3}

Step by step solution

01

- Isolate the variable

To solve for the variable, we need to isolate it on one side of the equation. The given equation is \(-\frac{3}{7} x = 4\). To isolate \(x\), we'll multiply both sides of the equation by the reciprocal of \(-\frac{3}{7}\), which is \(-\frac{7}{3}\).
02

- Multiply both sides by the reciprocal

Multiply both sides of the equation by \(-\frac{7}{3}\):\(-\frac{3}{7} x \times -\frac{7}{3} = 4 \times -\frac{7}{3}\).\This simplifies to: \(x = -\frac{28}{3}\).
03

- Simplify the solution

Now, simplify if necessary. In this case, \(x = -\frac{28}{3}\) is already in its simplest form.

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Key Concepts

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

Reciprocals
Understanding reciprocals is crucial when solving linear equations, especially when dealing with fractions. A reciprocal of a number is essentially what you multiply that number by to get 1. For example, the reciprocal of \(\frac{3}{7}\) is \(\frac{7}{3}\).
Let's see why:
  • When you multiply \(\frac{3}{7}\) and \(\frac{7}{3}\), the numerator of one fraction cancels out the denominator of the other, giving you 1: \[\frac{3}{7} \times \frac{7}{3} = 1\]
This rule helps a lot when you want to isolate a variable multiplied by a fraction.
Isolation of Variables
Isolation of variables is the process of getting the variable on one side of the equation by itself.
In the equation \-\frac{3}{7} x = 4\, we need to isolate \(x\).
To do this, follow these steps:
  • Recognize that \(x\) is being multiplied by \-\frac{3}{7}\.
  • The best way to undo this multiplication is by using its reciprocal, which will help us make the coefficient of \(x\) equal to 1.
Multiplying both sides by \-\frac{7}{3}\ allows us to cancel out the fraction on the left side:
\[ -\frac{3}{7} x \times -\frac{7}{3} = 4 \times -\frac{7}{3} \] Simplifying this gives us \(x = -\frac{28}{3}\).
This is a very efficient method for isolating variables in equations involving fractions.
Multiplication of Fractions
Multiplying fractions is a fundamental skill needed to solve equations like \-\frac{3}{7} x = 4\.
Here鈥檚 a quick refresher: When multiplying fractions, multiply the numerators together and the denominators together. For example:
  • \[ \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \]
Applying this to our problem, we need to multiply both sides of the equation by the reciprocal of \-\frac{3}{7}\, which is \-\frac{7}{3}\:
  • \[ -\frac{3}{7} x \times -\frac{7}{3} = 4 \times -\frac{7}{3} \]
On the left side, the numerators and denominators cancel each other out, leaving us with 1\(x\):
\[ x = -\frac{28}{3} \] Remember these multiply-and-cancel steps as they help to deal with fractions across many algebra problems.

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Most popular questions from this chapter

Show a complete solution to each problem. Highway miles. Berenice and Jarrett drive a rig for Continental Freightways. In 1 day Berenice averaged \(50 \mathrm{mph}\) and Jarrett averaged \(56 \mathrm{mph},\) but Berenice drove for 2 more hours than Jarrett. If together they covered 683 miles, then for how many hours did Berenice drive?

Solve each equation. Practice combining some steps. Look for more efficient ways to solve each equation. $$-\frac{2}{7} w=4$$

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Show a complete solution to each problem. Rectangular picture. If the perimeter of a rectangular picture is \(48 \mathrm{cm}\) and the width is \(3 \mathrm{cm}\) shorter than half the length, then what are the length and width?

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