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Chapter 2: Problem 8

Is 3 a solution of \(-4 x+1=-20\) ?

Short Answer

Expert verified
3 is not a solution because it does not satisfy the equation.

Step by step solution

01

Understand the problem

We need to determine if the number 3 satisfies the given equation: \[-4x + 1 = -20.\] This means we will substitute 3 for \(x\) and check to see if both sides of the equation are equal.
02

Substitute 3 into the equation

Replace \(x\) with 3 in the equation:\[-4(3) + 1.\]
03

Simplify the expression

Calculate \(-4 \times 3\) which gives \(-12\) and then add 1:\[-12 + 1 = -11.\]
04

Compare with the right side of the equation

Compare the result of the left side \(-11\) with the right side of the equation, which is \(-20\). Since \(-11 eq -20\), 3 is not a solution.

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Key Concepts

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

Solution Verification
When you're working with linear equations, verifying a solution is a key step to ensure your answer is correct. For any given equation, like \[ -4x + 1 = -20, \]you want to check if a potential solution actually satisfies the equation.Here's how you can verify a potential solution:
  • Substitute the proposed solution into the equation. For instance, if you're checking if 3 is a solution, replace every instance of \(x\) with 3.
  • Calculate the value of the equation's left side.
  • Compare it to the right side of the equation. If they match, you've found a valid solution. If not, your proposed solution isn't correct.
Solution verification ensures you're not just guessing and provides a reliable check of your work. It's a safety net for your calculations.
Substitution Method
The substitution method is an essential technique when solving equations, especially when you want to test if a number is a solution. In the given equation \[-4x + 1 = -20,\]we use substitution to see if 3 can solve for \(x\).Here鈥檚 how it works:
  • Take the equation and wherever you see \(x\), plug in the value you're testing鈥攊n this example, 3.
  • This turns the equation into an arithmetic problem that you can solve, for example, \[-4 imes 3 + 1.\]
  • If the result equals the right side of the original equation, then the number truly is a solution.
The substitution method is straightforward. It's used not only in solution verification but also in more complex systems of equations.
Simplifying Expressions
Simplifying expressions is crucial for solving equations efficiently. When we substitute a number into an equation, our goal is to make the computation simple.Take for example \[-4(3) + 1.\]The steps for simplifying are:
  • Perform multiplication first, as stated in the order of operations (PEMDAS/BODMAS). Here, calculating \(-4 imes 3 = -12.\)
  • Then, perform the addition or subtraction. So, add 1 to \(-12\), resulting in \(-12 + 1 = -11.\)
Simplifying ensures you鈥檙e working with the simplest form of an expression. This makes it easier to compare results against the original equation's right side. Mastering simplification helps keep your equations clear and your solutions accurate.

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