Problem 70 - Algebra 2

Question

Solve each equation. $$ 3 k+8=5 $$

Step-by-Step Solution

Answer

The solution is \(k = -1\).

Step 1: Isolate the Variable Term

Start by getting the term with the variable on one side by itself. Subtract 8 from both sides of the equation to move the constant term to the right. \[3k + 8 - 8 = 5 - 8\]This simplifies to:\[3k = -3\]

Step 2: Solve for the Variable

Now that we have isolated the term with the variable, solve for \(k\) by dividing both sides by 3 to find the value of \(k\):\[\frac{3k}{3} = \frac{-3}{3}\]Which simplifies to:\[k = -1\]

Key Concepts

Isolate the Variable

In the process of solving linear equations, it is crucial to "isolate the variable". This means getting the variable term on one side of the equation all by itself.
To do this, we must first eliminate any numbers or terms that are added or subtracted to the variable term.
In our example equation, \(3k + 8 = 5\), we begin by removing the constant term next to the variable, which is the +8.

• Subtract 8 from both sides of the equation: \(3k + 8 - 8 = 5 - 8\).
• This results in \(3k = -3\).

This step is about making the surrounding equation simpler, allowing a clear path to find the value of the variable. Once isolated, the variable term can be easily manipulated to solve the equation.

Inverse Operations

"Inverse operations" are fundamental tools used to solve equations. They are operations that undo each other, such as addition vs. subtraction or multiplication vs. division.
These operations help us move terms around and simplify equations by doing the opposite operation to both sides of the equation.

• If you add something to one side, subtract the same thing from both sides to maintain balance.
• If you multiply, divide both sides by the same value to undo the multiplication.

In the example equation, after isolating the variable term, we addressed the multiplication by 3:
By dividing both sides by 3, which undoes multiplication, we simplified the equation to \(k = -1\). Understanding inverse operations allows us to manipulate equations systematically.

Variable Term

The "variable term" refers to the part of an equation that includes the variable we are trying to solve for.
It usually appears in the form of a variable (like \(k\), \(x\), or \(y\)) multiplied by a number or between operators.
In our given problem, the variable term is \(3k\). This is because \(k\) is the unknown we want to isolate and solve for.

• The number 3 in \(3k\) is a coefficient, showing how many times the variable is counted.
• Our aim is to get \(k\) by itself to determine its value accurately.

Identifying and isolating the variable term is a crucial initial step in any equation-solving process, as it aligns our efforts specifically towards finding the unknown value.