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Chapter 2: Problem 129

Solve and check: \(-1+3(x-4)=2 x .\) (Section 1.2 , Example \(2)\)

Short Answer

Expert verified
The solution to the equation is \(x = 13\).

Step by step solution

01

Simplify the equation

Distribute 3 inside the parentheses on the left side of the equation, resulting in the new equation: \(-1 + 3x - 12 = 2x\). Further simplification leads to the equation: \(3x -13 = 2x\).
02

Isolate the variable x

Collect like terms by subtracting '2x' from both sides of our equation. This results in: \(x - 13 = 0\). Adding 13 to both sides then gives us: \(x = 13\).
03

Check the solution

Substitute \(x = 13\) back into the original equation. \(-1 + 3(13 - 4) = 3 * 9 = 27 - 1 = 26\) and \(2 * 13 = 26\). Since both sides match and are equal to 26, we can confirm that the solution to the equation is correct.

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Key Concepts

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

Understanding Linear Equations
A linear equation in one variable is an algebraic statement in which the highest power of the variable is one. These equations generally have a single solution. In the exercise \( -1+3(x-4)=2x \), the goal is to solve for the unknown variable \( x \). This equation represents a straight line when plotted on a graph.

The very essence of a linear equation is its simplicity and ease of manipulation, as seen through the step-by-step solution. By distributing and combining like terms, we obtain a simplified linear equation with variables on both sides, which is a common form of linear equations. Understanding how to handle such equations is fundamental in algebra, as it forms the basis for more complex mathematical concepts.
Isolating Variables
Isolating the variable is a key step in solving linear equations. It involves rearranging the equation to solve for the unknown variable by making it the subject of the formula. This often involves operations such as addition, subtraction, multiplication, and division. In our example, isolating \( x \) helped us find the solution.

Tips for Isolating Variables

  • Maintain balance by performing the same operation on both sides of the equation.
  • Combine like terms to simplify the equation.
  • Knowing the properties of equality is useful in these manipulations.
Grasping this concept gives you the ability to crack open the hidden value of the variable in any linear equation.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and arithmetic operations. In the context of our exercise, \( 3(x-4) \) is an algebraic expression that we break down into simpler parts by distribution.

An understanding of algebraic expressions is essential since they are the very language through which we articulate mathematical problems and solutions. When working with algebraic expressions, it's important to:
  • Identify like terms for simplification.
  • Understand the significance of each arithmetic operation within the expression.
  • Recognize the order of operations (PEMDAS/BODMAS) to correctly simplify or manipulate expressions.
Mastering algebraic expressions is like obtaining a key to unlocking a vast array of mathematical puzzles.

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Most popular questions from this chapter

Find a. \((f \circ g)(x)\) b. \((g \circ f)(x)\) c. \((f \circ g)(2)\) d. \((g \circ f)(2)\) $$f(x)=\frac{2}{x}, g(x)=\frac{2}{x}$$

graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$\begin{array}{r} {x^{2}+y^{2}=9} \\ {x-y=3} \end{array}$$

The regular price of a pair of jeans is \(x\) dollars. Let \(f(x)=x-5\) and \(g(x)=0.6 x\) a. Describe what functions \(f\) and \(g\) model in terms of the price of the jeans. b. Find \((f \circ g)(x)\) and describe what this models in terms of the price of the jeans. c. Repeat part (b) for \((g \circ f)(x)\) d. Which composite function models the greater discount on the jeans, \(f \circ g\) or \(g \circ f ?\) Explain.

graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$\begin{array}{r} {x^{2}+y^{2}=16} \\ {x-y=4} \end{array}$$

Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$x^{2}+y^{2}+3 x+5 y+\frac{9}{4}=0$$
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