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Chapter 1: Problem 69

Solve the equation. $$18 \mathrm{x}=43$$

Short Answer

Expert verified
\(x = 43/18\)

Step by step solution

01

Identify the Coefficients and Constants

In this problem, \(18\) is the coefficient, which is multiplying the variable \(x\), and \(43\) is the constant.
02

Divide Both Sides by the Coefficient

To isolate \(x\), we will divide both sides of the equation \(18x = 43\) by \(18\), resulting in \(x = 43/18\).

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Key Concepts

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

Isolate variable
When it comes to solving linear equations, a fundamental goal is to isolate the variable. This simply means that you want to get the variable by itself on one side of the equation, with no other numbers or variables attached to it.

To achieve this, you perform a series of operations that 'undo' whatever is being done to the variable. In the given exercise, the variable, represented by the letter 'x', is being multiplied by 18. To 'undo' this multiplication, we must perform the opposite operation, which in this case is division. Isolating the variable is the key to finding its value, which is the solution to the equation.
Coefficients in algebra
In algebra, the terms 'coefficients' and 'variables' are intertwined. A coefficient is a number used to multiply a variable. In the equation from the exercise, 18 is the coefficient of the variable x. Algebraic expressions and equations frequently consist of several terms with coefficients.

Understanding coefficients is crucial as it helps you identify the strength of the relationship between the variable and its outcome鈥攊n other words, how much the variable is 'worth' in the equation. Being able to identify and manipulate coefficients is a foundational skill in algebraic manipulation.
Division in equations
Division is one of the four basic operations in mathematics and plays a significant role in solving equations. It is used to reverse multiplication. If a variable is multiplied by a number (the coefficient), you can divide both sides of the equation by that number to 'cancel' it out and get one step closer to isolating the variable.

In our example, we divide both sides of the equation 18x = 43 by 18. This division leads to the result x = 43/18. It's crucial to remember that what you do to one side of the equation, you must do to the other to keep the equation balanced.
Algebraic manipulation
Algebraic manipulation involves rearranging and simplifying algebraic expressions and equations. It encompasses various techniques, including adding, subtracting, multiplying, dividing, factoring, and expanding.

In the context of our exercise, the manipulation involved dividing both sides of the equation by the coefficient of the variable to isolate 'x'. Mastering algebraic manipulation allows students to tackle more complex equations and systems of equations, as they provide the tools to rearrange and solve for unknowns effectively.

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Most popular questions from this chapter

In Exercises \(56-63,\) complete the statement with always, sometimes, or never. Explain your reasoning. A line ______ has endpoints.

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