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Chapter 5: Problem 302

In the following exercises, solve the equation. $$ k-3.19=-4.6 $$

Short Answer

Expert verified
k = -1.41

Step by step solution

01

Identify the equation

The given equation is \( k - 3.19 = -4.6 \)
02

Isolate the variable

To solve for \( k \), add 3.19 to both sides of the equation: \( k - 3.19 + 3.19 = -4.6 + 3.19 \)
03

Simplify both sides

Simplify the left side of the equation, which eliminates the \( -3.19 \) \( k = -4.6 + 3.19 \) Now simplify the right-hand side: \( -4.6 + 3.19 = -1.41 \)
04

Write the final solution

The solution is \( k = -1.41 \)

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Key Concepts

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

Isolate the Variable
When solving equations, our first goal is to isolate the variable. This means we want the variable (in this case, \(k\)) to be by itself on one side of the equation. Isolating the variable helps us find its value
To do this, we often need to undo whatever operations are being applied to the variable. In our example, the equation is \(k - 3.19 = -4.6\). Here, 3.19 is being subtracted from \(k\).
So, to isolate \(k\), we need to do the opposite of subtraction, which is addition.
We add 3.19 to both sides of the equation. Doing the same operation on both sides keeps the equality balanced, which is crucial in solving equations.
Simplify Equations
Once you've isolated the variable, the next step is to simplify the equation. Simplification means performing all possible operations so that we get the variable alone with its value.
In our example, after adding 3.19 to both sides, we get:
\[ k - 3.19 + 3.19 = -4.6 + 3.19 \]
On the left side of the equation, \(-3.19\) and \(+3.19\) cancel each other out, leaving us with just \(k\).
On the right side, we need to perform the arithmetic:
\[ -4.6 + 3.19 = -1.41 \]
This step simplifies the equation to \(k = -1.41\). Always double-check your arithmetic to make sure you've got the correct simplified form of the equation.
Linear Equations
The equation from our example, \(k - 3.19 = -4.6\), is a linear equation. Linear equations are equations of the first degree, meaning the variable (in this case, \(k\)) is not raised to any power other than one. They form a straight line when graphed.
Solving linear equations involves a series of steps:
  • Identify the equation
  • Isolate the variable
  • Simplify both sides
  • Write the final solution

In our specific example, after simplification and isolating the variable, we found the solution to be \(k = -1.41\). Remember, knowing these steps will help you tackle any linear equation you come across in your studies.

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