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Chapter 7: Problem 69

Solve the following linear and quadratic equations. See Sections 2.3 and 6.6 $$ 3 x+5=7 $$

Short Answer

Expert verified
The solution is \( x = \frac{2}{3} \).

Step by step solution

01

Identify the Type of Equation

The equation given is a linear equation of the form \( ax + b = c \). It involves a variable raised to the first power, which makes it linear.
02

Isolate the Variable

Subtract 5 from both sides of the equation to isolate the term with the variable. This gives:\[ 3x + 5 - 5 = 7 - 5 \]which simplifies to:\[ 3x = 2 \]
03

Solve for the Variable

Divide both sides of the equation by 3 to solve for \( x \):\[ x = \frac{2}{3} \]

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Key Concepts

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

Solving Linear Equations
Solving linear equations is a fundamental concept in algebra. A linear equation is one where the highest power of the variable is one, which makes it quite straightforward to solve.
The basic form of a linear equation is usually represented as \( ax + b = c \). Here, \( a \), \( b \), and \( c \) are constants, while \( x \) is the variable you need to solve for.
The goal is to manipulate the equation so that you can find the exact value of \( x \) that makes the equation true.
  • Start by simplifying the equation if necessary, such as combining like terms.
  • Then follow the steps to isolate the variable and solve.
These steps often involve undoing operations, like addition or multiplication, that are applied to the variable.
Isolating the Variable
To solve an equation, isolating the variable is crucial. This means you want the variable by itself on one side of the equation.
In the equation \( 3x + 5 = 7 \) shown, our aim is to make \( x \) alone on one side. Here's a simple strategy:
  • Identify operations performed on the variable (like addition or multiplication).
  • Perform the opposite operation to both sides of the equation. In this case, subtract 5 from both sides to eliminate it from the left side: \( 3x + 5 - 5 = 7 - 5 \) simplifies to \( 3x = 2 \).
This step-by-step approach helps set the stage to solve for the variable easily.
The Power of One in Equations
When an equation involves a variable raised to the first power, it's called a linear equation.
The first power characteristic means each term with the variable simply has it multiplied by some constant, or just the variable itself.
In the example \( 3x + 5 = 7 \), the term \( 3x \) is the main linear component.
  • Linear equations are straightforward since they do not involve squares or higher powers, which can complicate the solution process.
  • They typically have one solution, making the process quick and direct.
Understanding the power of one is key to recognizing and solving these types of equations quickly.
Basic Algebra Concepts
Basic algebra concepts form the foundation for solving a variety of equations. Understanding these builds confidence in tackling math problems.
Among the core ideas are operations such as:
  • Addition and Subtraction: Used to move terms across the equation to help isolate the variable.
  • Multiplication and Division: Often used in the final steps to find the solution for the variable after it's isolated.
These operations are reversible, which is key to manipulating equations.
Remember, in solving equations, the process is all about maintaining balance between the two sides of the equation. Keep the equation equal by doing operations equally on both sides.

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Most popular questions from this chapter

Solve. See the Concept Check in the Section. Which of the following are equivalent to \(\frac{\frac{a}{7}}{\frac{b}{13}} ?\) a. \(\frac{a}{7} \cdot \frac{b}{13}\) b. \(\frac{a}{7} \div \frac{b}{13}\) c. \(\frac{a}{7} \div \frac{13}{b}\) d. \(\frac{a}{7} \cdot \frac{13}{b}\)

During a storm, water treatment engineers monitor how quickly rain is falling. If too much rain comes too fast, there is a danger of sewers backing up. A formula that gives the rainfall intensity \(i\) in millimeters per hour for a certain strength storm in eastern Virginia is $$ i=\frac{5840}{t+29} $$ where \(t\) is the duration of the storm in minutes. What rainfall intensity should engineers expect for a storm of this strength in eastern Virginia that lasts for 80 minutes? Round your answer to one decimal place.

\- Solve the following. See Examples I through 7. (Note: Some exercises can be modeled by equations without rational expressions.) A pilot can fly an MD-11 2160 miles with the wind in the same time as she can fly 1920 miles against the wind. If the speed of the wind is \(30 \mathrm{mph}\), find the speed of the plane in still air. (Source: Air Transport Association of America)

Perform each indicated operation. In ice hockey, penalty killing percentage is a statistic calculated as \(1-\frac{G}{P},\) where \(G=\) opponent's power play goals and \(P=\) opponent's power play opportunities. Simplify this expression.

Perform each indicated operation. A board of length \(\frac{3}{x+4}\) inches was cut into two pieces. If one piece is \(\frac{1}{x-4}\) inches, express the length of the other piece as a rational expression. IMAGE CANNOT COPY!
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