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Chapter 7: Problem 79

In Exercises \(76-79,\) solve for \(x\) $$4(x-1)-3 x=x-4$$

Short Answer

Expert verified
The solution is all real numbers.

Step by step solution

01

- Distribute the 4

Apply the distributive property to the left side of the equation to eliminate the parentheses. This means multiplying 4 by both terms inside the parentheses: 4(x-1) - 3x = x - 4 becomes: 4x - 4 - 3x = x - 4
02

- Combine like terms

Combine the like terms on the left side of the equation. The terms 4x and -3x are like terms: 4x - 3x - 4 = x - 4 This simplifies to: x - 4 = x - 4
03

- Isolate the variable

In this case, subtract x from both sides to isolate the remaining terms: x - x - 4 = x - x - 4 This simplifies to: -4 = -4
04

- Analyze the resulting statement

The resulting equation -4 = -4 is always true, which means the original equation is an identity. This implies that the solution to the equation is all real numbers. Therefore, any real number for x will satisfy the equation.

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Key Concepts

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

distributive property
The distributive property is a fundamental concept used to simplify expressions and equations in algebra. It allows you to remove parentheses by distributing (multiplying) a term across the terms inside the parentheses. For example, in the exercise, we started with the equation: $$4(x-1) - 3x = x - 4$$. Using the distributive property means we multiply 4 by both terms in the parentheses: (4 \times x) and (4 \times -1). This changes our equation to: $$4x - 4 - 3x = x - 4$$. Remember, the goal here is to eliminate the parentheses to make solving the equation easier.
combining like terms
Combining like terms is another essential algebraic technique. Like terms are terms that have the same variable raised to the same power. In the equation: $$4x - 4 - 3x = x - 4$$, the terms '4x' and '-3x' are like terms because they both contain the variable 'x'. By combining them, we subtract 3x from 4x to get: $$4x - 3x - 4 = x - 4$$. This simplifies to: $$x - 4 = x - 4$$. Simplifying like terms helps in reducing complex equations to simpler forms, making it easier to solve for the variable.
identity equations
Identity equations are equations that are true for all values of the variable. After combining like terms and simplifying further, we got: $$x - 4 = x - 4$$. When we try to isolate the variable 'x' by subtracting 'x' from both sides, we end up with: $$-4 = -4$$. This statement is always true, no matter what value of 'x' we choose. Hence, it is an identity equation. This means any real number substituted for 'x' will satisfy the original equation. Understanding identity equations is crucial because it tells us that there is no single solution but infinitely many solutions for the variable.

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