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In Mathematics / College | 2025-07-05

Which of the following statements completely describes the solution set for [tex]8(x-13)=8 x-104[/tex]?
A. There is no solution for x
B. [tex]x=13[/tex]
C. [tex]x=0[/tex] only
D. All real numbers are solutions for x
E. [tex]x=8[/tex] only

Asked by b5mx49vcs8

Answer (2)

Expand the left side of the equation: 8 ( x − 13 ) = 8 x − 104 .
Simplify the equation: 8 x − 104 = 8 x − 104 .
Subtract 8 x from both sides: − 104 = − 104 .
Since the equation is always true, all real numbers are solutions: All real numbers are solutions for x ​ .

Explanation

Analyze the problem We are given the equation 8 ( x − 13 ) = 8 x − 104 and asked to find the solution set for x . This means we need to determine which values of x , if any, satisfy the equation. We will start by expanding the left side of the equation.

Expand the equation Expanding the left side of the equation, we have: 8 ( x − 13 ) = 8 x − 8 × 13 We know that 8 × 13 = 104 , so the equation becomes: 8 x − 104 = 8 x − 104

Simplify the equation Now, we want to isolate x on one side of the equation. We can subtract 8 x from both sides: 8 x − 104 − 8 x = 8 x − 104 − 8 x − 104 = − 104

Determine the solution set Since − 104 = − 104 is always true, regardless of the value of x , this means that the equation is an identity. In other words, any real number substituted for x will satisfy the equation. Therefore, all real numbers are solutions for x .

State the final answer The solution set for the equation 8 ( x − 13 ) = 8 x − 104 is all real numbers.


Examples
Understanding equations like this is crucial in many real-world scenarios. For instance, imagine you're managing a budget where expenses are directly proportional to the number of items purchased. If the equation holds true for all quantities, it means your budget is perfectly balanced, and there's no surplus or deficit regardless of how many items you buy. This concept is also applicable in physics, where equations describe relationships between variables, and an identity implies a fundamental law holds true under all conditions.

Answered by GinnyAnswer | 2025-07-05

The equation 8 ( x − 13 ) = 8 x − 104 simplifies to an identity, meaning − 104 = − 104 holds true for all values of x . Therefore, the solution set includes all real numbers. The correct answer is D: All real numbers are solutions for x .
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Answered by Anonymous | 2025-07-18

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