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Chapter 5: Problem 61

Determine whether each equation is true or false if \(a=2\) \(b=5,\) and \(c=6.\) $$2 a b=20$$

Short Answer

Expert verified
True, the equation is true for the given values.

Step by step solution

01

Substitute Variables

First, substitute the given values for the variables into the equation. The original equation is \(2ab = 20\). With \(a = 2\) and \(b = 5\), substitute them into the equation to get \(2 \times 2 \times 5\).
02

Calculate Left Side

Calculate the left side of the equation after substitution. This involves multiplying the numbers: \(2 \times 2 = 4\) and then \(4 \times 5 = 20\).
03

Compare Both Sides

Compare the calculated left side of the equation with the right side to determine if the equation is true. In this case, both sides of the equation are equal to 20.
04

Conclusion

Based on the comparison, the equation \(2ab = 20\) is true when \(a=2\) and \(b=5\).

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Key Concepts

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

Equation Solving
Equation solving is like a puzzle where you have to find the missing pieces to complete the picture. In mathematics, an equation is a statement that asserts the equality of two expressions. For example, in our problem, we have the equation \(2ab = 20\). To solve this puzzle, we need to check if, when we put numbers in place of the letters, both sides remain equal. The main goal here is to see if the left expression, \(2ab\), equals the numeric value presented on the right side, which is 20. By comparing both sides after calculating them with substituted values, we determine if the equation holds true or is false.
Variable Substitution
Variable substitution is the process of replacing variables in an equation with given values. In the original exercise, we were provided with specific values for \(a\) and \(b\): that is \(a=2\) and \(b=5\). When you substitute, you just plug in these values into the equation. So, the equation \(2ab = 20\) becomes \(2 \times 2 \times 5\). Substitution is essential because it transforms the equation from being abstract to something tangible we can manipulate and solve. By practicing substitution, you become better at handling more complex equations as well, making it a foundational skill in algebra.
Truth Values of Equations
The truth value of an equation tells us whether the equation is correct or not, based on the given values. An equation is deemed "true" if, after substitution and calculation, both sides are equal. Conversely, it is "false" if they are not. In our example, after substituting values and calculating, both sides of the equation equal 20, confirming it is true. This process is crucial in algebra because it tests our understanding and ability to apply mathematical principles. Finding truth in equations builds a deeper understanding of how numbers interact with each other in predictable ways.

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