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Chapter 2: Problem 2

Is the given value a solution to the linear equation? $$ y+7=-6 ; \quad y=-13 $$

Short Answer

Expert verified
No, \(x = -1\) is not a solution to the equation.

Step by step solution

01

Substitute the Value

Replace every occurrence of \( x \) in the equation with the given value \( x = -1 \). The equation becomes: \[ -(-1) + 17 - 8(-1) = 9 - (-1) \].
02

Simplify Both Sides

Simplify the expressions on both sides of the equation. For the left side: \[ -(-1) + 17 - 8(-1) \] becomes -(-1) = 1, -8(-1) = 8, so \[ 1 + 17 + 8 = 26 \].For the right side: \[ 9 - (-1) \] becomes \[ 9 + 1 = 10 \].
03

Compare Both Sides

Compare the simplified expressions from both sides of the original equation. The left side simplifies to 26, and the right side simplifies to 10. Since 26 is not equal to 10, \( x = -1 \) is not a solution to the equation.

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Key Concepts

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

Solution Verification
When we solve a linear equation, one of the crucial steps is to verify if a particular value is a solution. Solution verification helps ensure accuracy in problem-solving by confirming our calculations. To check a solution, we must substitute the suggested value for the variable back into the original equation.
  • Substitute the given value for every occurrence of the variable.
  • Simplify each side of the equation separately to revealing their simplified forms.
  • Finally, compare these values. If both sides are equal, the value is a valid solution; if not, it isn't.
In the original exercise, the value \( x = -1 \) was substituted into the equation. After simplifying both sides, we found 26 on the left and 10 on the right. Thus, these differing results demonstrate that \( x = -1 \) does not satisfy the equation, verifying it is not a solution.
Algebraic Substitution
Algebraic substitution is the method of replacing variables with given numbers or expressions to evaluate or simplify equations. It is a fundamental operation in algebra and plays a pivotal role in verifying solutions.The approach involves these steps:
  • Identify the variable to be replaced in the equation.
  • Substitute the variable with its given value, paying extra attention to keeping expressions intact.
  • Perform necessary calculations to simplify the results.
In our problem, by substituting \( x = -1 \) into every place \( x \) appears in the equation, we ensured all occurrences of the variable were consistently replaced, setting the stage for effective simplification.
Equation Simplification
Equation simplification reduces complex expressions into their simplest forms. It is essential for understanding equations and making accurate comparisons.Here's how simplification works in practice:
  • Perform arithmetic operations like adding, subtracting, multiplying, or dividing numbers, within parentheses where necessary.
  • Simplify signed numbers and perform operations respecting signs, converting subtraction of a negative into addition.
  • Reformulate expressions to their simplest terms to facilitate analysis.
For our given linear equation, both sides of the equation were simplified: - On the left, combing terms like \(-(-1) = 1\) and \(-8(-1) = 8\), which resulted in 26. - Meanwhile, the right simplified to 10, which clearly does not match with the left.This simplification is critical for easy comparison, confirming whether a solution is valid.

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Most popular questions from this chapter

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ -15 \leq 23 x-15<45 $$

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