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Chapter 3: Problem 67

Is \(-2.8\) a solution of the equation \(0.8-p=3.6 ?\)

Short Answer

Expert verified
Yes, \(-2.8\) is a solution to the equation \(0.8 - p = 3.6\).

Step by step solution

01

Identify the proposed solution

The proposed solution for the equation is stated as \(-2.8\). That means we are examining if \(p=-2.8\) is a solution to the given equation \(0.8-p=3.6\).
02

Substitute the proposed solution

We substitute \(p=-2.8\) into the original equation in place of \(p\). Doing this, we get: \(0.8 - (-2.8) = 3.6\).
03

Evaluate the equation

Next, simplify the left side of the equation: \(0.8 - (-2.8)\) equals \(3.6\), so the equation simplifies to \(3.6 = 3.6\)
04

Check solution

Since \(3.6 = 3.6\) is a true statement, it means that \(-2.8\) is indeed a solution to the original equation.

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Key Concepts

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

Substitution Method
The substitution method is a helpful technique often used to test if a particular number is a solution to an equation. To do this, you replace the variable in the equation with the given number.

In our exercise, the variable is \( p \) and we want to know if \(-2.8\) fits into the equation \( 0.8 - p = 3.6 \). We substitute \(-2.8\) for \( p \), changing the original equation to \( 0.8 - (-2.8) = 3.6 \).

This substitution helps us focus on whether the left side of the equation will equal the right side once the number is plugged in. It's a direct way to test if an equation holds true for a given value.
Simplifying Equations
Simplifying equations is a critical step in solving them, especially when using the substitution method. In our example, after substituting \(-2.8\) into the equation, we get \( 0.8 - (-2.8) \).

This step requires us to recognize that subtracting a negative is the same as adding the positive equivalent:
  • \(0.8 - (-2.8)\) becomes \( 0.8 + 2.8 \)
  • This simplifies to \( 3.6 \)

Simplifying makes it easier to see if both sides of the equation are equal. Here, \(3.6 = 3.6\) confirms the equation holds true after substitution.
Checking Solutions
Checking solutions is the final step that confirms whether a substituted value is truly a solution to the equation. After you simplify both sides using the substitution method, compare them.

In this example, the left side \(3.6\) exactly equals the right side \(3.6\). This equality indicates that the original proposal, \(-2.8\) as a solution, is correct.
Checking the equality is essential as it gives confidence that the solution is valid. If the sides didn't match, \(-2.8\) wouldn't be the solution, prompting further investigation.

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Most popular questions from this chapter

On January \(22,1943,\) in Spearfish, South Dakota, the temperature fell from \(12.22^{\circ} \mathrm{C}\) at 9 A.M. to \(-20^{\circ} \mathrm{C}\) at 9: 27 A.M. How many degrees did the temperature fall during the 27 -minute period?

Given that \(x\) is an integer, find all values of \(x\) for which \(|x|<5\)

What is 6 less than \(-9 ?\)

Subtract. $$-4-(-4)$$

a. Name two numbers that are 4 units from 2 on the number line. b. Name two numbers that are 5 units from 3 on the number line.
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