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Chapter 5: Problem 301

In the following exercises, solve the equation. $$ j-1.82=-6.5 $$

Short Answer

Expert verified
j = -4.68

Step by step solution

01

- Understand the Equation

The given equation is: \[ j - 1.82 = -6.5 \]To solve for \(j\), isolate \(j\) on one side of the equation.
02

- Add 1.82 to Both Sides

To isolate \(j\), add 1.82 to both sides of the equation:\[ j - 1.82 + 1.82 = -6.5 + 1.82 \]This simplifies to:\[ j = -6.5 + 1.82 \]
03

- Calculate the Right-Hand Side

Now calculate the value on the right-hand side:\[ -6.5 + 1.82 = -4.68 \]Thus:\[ j = -4.68 \]

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Key Concepts

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

Isolating Variables
In algebra, isolating the variable is a fundamental step when solving equations.
It means getting the variable by itself on one side of the equation. This helps in determining its value.
Let鈥檚 look at the given equation: \( j - 1.82 = -6.5 \)
To isolate \( j \), we need to eliminate the other term, \( -1.82 \).
A straightforward way to do this is by adding \( 1.82 \) to both sides.
Why add \( 1.82 \)? Because it will cancel out the \( -1.82 \) on the left side, leaving \( j \) by itself.
Thus, isolating the variable is a crucial step to make the equation simpler and get the desired answer.
Addition Property of Equality
The Addition Property of Equality states that you can add the same number to both sides of an equation without changing the equality.
Using our equation: \( j - 1.82 = -6.5 \)
To remove \( -1.82 \) from the left side, we add \( 1.82 \) to both sides:\( j - 1.82 + 1.82 = -6.5 + 1.82 \)
This property ensures that the equation remains balanced after the addition.
It is a very useful tool in algebra for isolating variables.
In essence, if you do something to one side of the equation, do the same to the other side to keep the equation true.
Simplification in Algebra
After applying the addition property of equality, we need to simplify the equation.
We started with:\( j - 1.82 + 1.82 = -6.5 + 1.82 \)
Upon simplifying, the left side becomes \( j \), leaving:\( j = -6.5 + 1.82 \)
Now, we perform the arithmetic operation on the right side:\( -6.5 + 1.82 = -4.68 \)
Therefore, \( j = -4.68 \).
Simplification involves combining like terms or performing basic arithmetic to make the equation straightforward and easier to understand.
Always double-check your arithmetic to ensure accuracy.

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