Problem 24 - Prealgebra

Question

Solve equation. Check your solution. \(12+1.5 a=3 a\)

Step-by-Step Solution

Answer

The solution is \(a = 8\), and it satisfies the original equation.

Step 1: Isolate Variable Term

Start by isolating the term with the variable on one side. To do this, subtract \(1.5a\) from both sides of the equation:\[12 + 1.5a - 1.5a = 3a - 1.5a\]This simplifies to:\[12 = 1.5a\]

Step 2: Solve for the Variable

Now that the equation is simplified, solve for \(a\) by dividing both sides by \(1.5\):\[\frac{12}{1.5} = \frac{1.5a}{1.5}\]Simplifying, we find:\[8 = a\]

Step 3: Check the Solution

Substitute \(a = 8\) back into the original equation to verify the solution:\[12 + 1.5(8) = 3(8)\]Calculate each side:\[12 + 12 = 24\]\[24 = 24\]Since both sides of the equation are equal, the solution is correct.

Key Concepts

Solving Linear Equations

Solving linear equations is the process of finding the value of the unknown variable that satisfies the equation. A linear equation generally takes the form: \[ ax + b = cx + d \]where \(a\), \(b\), \(c\), and \(d\) are constants, and \(x\) is the unknown variable.
To solve such equations efficiently, the goal is to manipulate the equation to isolate the variable on one side of the equation. This involves various algebraic operations such as addition, subtraction, multiplication, and division.
Let's consider the example: \[ 12 + 1.5a = 3a \]Here, the goal is to achieve a format where \(a\) is alone on one side to find its value.

Isolation of Variable

Isolating the variable is a fundamental step in solving equations. This means transforming the equation so that the variable we are solving for is on one side and everything else is on the other.
For the equation \[ 12 + 1.5a = 3a \],the first step is to remove \(1.5a\) from the left side. We do this by subtracting \(1.5a\) from both sides: \[ 12 + 1.5a - 1.5a = 3a - 1.5a \]Reducing this simplifies it to: \[ 12 = 1.5a \]. Here, the term with the variable is now isolated, making it easier to solve. This simplification is crucial, as it effectively sets up a straightforward algebraic operation to find the value of \(a\).

Checking Solutions

After solving for the variable, it's important to verify the solution by substituting it back into the original equation. This ensures the solution is accurate. For \[ a = 8 \]as found from the simplified equation \[ 12 = 1.5a \],substitute \(a = 8\) back into the initial equation \[ 12 + 1.5(8) = 3(8) \].
Upon calculation, both sides equal 24, confirming the accuracy of the solution. This step demonstrates that the found value holds true to the conditions set by the original equation. Ensuring the left and right sides of the equation remain balanced solidifies the correctness of the solution, providing complete confidence in your result.