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Chapter 2: Problem 42

Determine whether the given number is a solution of the equation. $$\frac{c}{-5}=-2 ;-10$$

Short Answer

Expert verified
-10 is not a solution for the given equation

Step by step solution

01

Substitute the Given Value into the Equation

Replace \(c\) with -10 in the equation. This gives us \( \frac{-10}{-5} = -2 \)
02

Simplify

Calculate the left-hand side of the equation. The division of -10 by -5 gives 2. This results in the equation 2 = -2
03

Compare both Sides

Check if the equation holds. The left side is 2 whereas right side is -2 so the equation doesn't hold true.

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Key Concepts

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

Algebra
Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. In this context, the symbols represent numbers, and they form equations and expressions that we can solve. Algebra is like a puzzle where the letters stand in for quantities we don't know yet.
  • When working with algebra, the most important thing is to understand the operation rules, such as addition, subtraction, multiplication, and division.
  • Equations are statements that show two sides being equal. They often include a variable like \(c\), which is an unknown value we are solving for.
Being comfortable with these basic concepts allows you to solve for unknowns and verify if certain values satisfy the given algebraic equation.
Substitution Method
The substitution method is a technique used for solving equations by replacing a variable with a number. In the given problem, we were tasked with determining if \(-10\) is a solution to the equation \(\frac{c}{-5} = -2\).

  • First, replace the variable \(c\) in the equation with \(-10\), converting the algebraic expression into a numerical one: \(\frac{-10}{-5}\).
  • Next, simplify the numerical expression by performing the division: divide \(-10\) by \(-5\) to get \(2\).
The substitution method helps in transforming an algebraic equation into a simpler form, which makes it easier to evaluate and solve for unknowns.
Solution Verification
Solution verification is an essential step to confirm whether the substituted value satisfies the original equation. Here's how it works:
  • Once you substitute the given value into the equation and simplify, check if the result on the left-hand side equals the right-hand side of the equation.
  • In our example, after substitution and simplification, we got \(2\) on the left-hand side, which we then compared to the right-hand side \(-2\).
  • Since \(2\) does not equal \(-2\), \(-10\) is not a solution to the equation \(\frac{c}{-5} = -2\).
Verification ensures that the steps taken to solve the equation are correct and confirms the validity of the proposed solution. If the sides are not equal after substitution, it means the given number is not a solution.

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Most popular questions from this chapter

A store sells regular coffee for \(7\)dollar a pound and a gourmet coffee for \(12\)dollar a pound. To reduce the inventory by 30 pounds of the gourmet coffee, the shopkeeper plans to make a gourmet blend to sell for \(9\)dollar a pound. How many pounds of regular coffee should be used?

Solve each inequality and graph the solution on the number line. $$1 \leq-7 x+8 \leq 15$$

Solve each application involving percents. Sales tax on a \(\$ 12\) compact disc is \(\$ 0.72 .\) At what rate is sales tax computed?

Find the solution of \(1 < \frac{1}{x} .\) (Hint: Will any negative values of \(x\) work?)

Solve each inequality and graph the solution on the number line. $$\frac{3(x-1)}{4} > x+1$$
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