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Chapter 9: Problem 67

Solve each equation for \(x .\) See Sections \(2.1,2.3,\) and 5.8 $$ 6 x-3(2-5 x)=6 $$

Short Answer

Expert verified
The solution is \(x = \frac{4}{7}\).

Step by step solution

01

Distribute the -3

First, expand the expression by multiplying -3 with each term inside the parentheses in the equation \(6x - 3(2 - 5x) = 6\). This results in \(6x - 6 + 15x = 6\).
02

Combine like terms

Combine the \(x\) terms on the left-hand side of the equation: \(6x + 15x\) gives \(21x\). So the equation becomes \(21x - 6 = 6\).
03

Isolate the 21x term

Add 6 to both sides of the equation to get \(21x = 12\).
04

Solve for x

Divide both sides by 21 to isolate \(x\). This gives \(x = \frac{12}{21}\). Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor, which is 3, to get \(x = \frac{4}{7}\).

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Key Concepts

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

Equation Solving
Equation solving is like being a detective. We need to find the value of the unknown variable, usually represented by \( x \). To do this, we perform various mathematical operations to isolate \( x \) on one side of the equation.
This involves careful and logical steps, such as using the distributive property, combining like terms, and simplifying fractions. Patience and attention to detail are key when solving equations.
In this exercise, our task was to solve the equation \( 6x - 3(2 - 5x) = 6 \) through various operations, eventually leading to the answer \( x = \frac{4}{7} \). Let's dive deeper into each step to understand how we get there.
Distributive Property
The distributive property is a useful tool in algebra. It allows us to multiply a number across terms within parentheses.
For instance, in the expression \(-3(2 - 5x)\), we distribute \(-3\) to both \(2\) and \(-5x\).
  • This means \(-3 \times 2 = -6\)
  • And \(-3 \times -5x = 15x\).
Replacing \(-3(2 - 5x)\) with \(-6 + 15x\) gives us \(6x - 6 + 15x = 6\). Understanding this property is crucial for simplifying and solving equations.
Combining Like Terms
Once the distributive property is applied, the next step is to simplify the expression by combining like terms. These are terms that have the same variable parts.
In our equation, we have \(6x\) and \(15x\) on the left-hand side. Both terms contain the variable \(x\).
  • Adding these together gives us \(21x\).
This step simplifies the equation, turning it into \(21x - 6 = 6\). Combining like terms helps to reduce the number of terms and makes the equation easier to solve.
Fraction Simplification
Finally, we simplify any fractions to get the most straightforward solution. When dividing both sides of the equation \(21x = 12\) by \(21\), you get \(x = \frac{12}{21}\).
The fraction can be simplified by dividing the numerator and the denominator by their greatest common divisor.
  • For \(12\) and \(21\), the greatest common divisor is \(3\).
  • Dividing both by \(3\) gives \(x = \frac{4}{7}\).
Simple fractions are clearer and more manageable, representing the solution in its simplest form. This step is essential to ensure accuracy and ease of interpretation in math problems.

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Most popular questions from this chapter

Explain why negative numbers are not included as logarithmic bases.

Simplify each rational expression. See Section 6.1. $$ \frac{x-5}{5-x} $$

Graph each function by finding ordered pair solutions, plotting the solutions, and then drawing a smooth curve through the plotted points. $$ f(x)=e^{-3 x} $$

Graph each function by finding ordered pair solutions, plotting the solutions, and then drawing a smooth curve through the plotted points. $$ f(x)=-2 \log x $$

Graph each function by finding ordered pair solutions, plotting the solutions, and then drawing a smooth curve through the plotted points. $$ f(x)=3 e^{x} $$
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