Problem 18 - Prealgebra

Question

Solve each equation. Check your solution. $$3 j-9=12$$

Step-by-Step Solution

Answer

The solution to the equation is \(j = 7\).

Step 1: Add 9 to both sides

The first step is to isolate the term with the variable on one side of the equation. Start by adding 9 to both sides of the equation to eliminate the -9. Equation: \[3j - 9 + 9 = 12 + 9\]This simplifies to:\[3j = 21\]

Step 2: Divide both sides by 3

Now that we have isolated the term with the variable, divide both sides of the equation by 3 to solve for \(j\).Equation: \[\frac{3j}{3} = \frac{21}{3}\]This simplifies to:\[j = 7\]

Step 3: Check the solution

To verify the solution, substitute \(j = 7\) back into the original equation and check if both sides are equal.Original equation:\[3j - 9 = 12\]Substitute \(j = 7\):\[3(7) - 9 = 12\]Calculate:\[21 - 9 = 12\]This simplifies to\[12 = 12\]Since both sides are equal, the solution \(j = 7\) is correct.

Key Concepts

Isolation of Variables

When solving equations, one of the primary goals is to isolate the variable. This means getting the variable on its own, on one side of the equation. It's an essential step in finding the value that makes the equation true. To do this, you need to perform mathematical operations that simplify the equation.
Let's dive into an example: consider the equation \(3j - 9 = 12\). Here, the variable is \(j\), and our task is to isolate it.

• First, identify the terms on the same side as the variable that we want to eliminate. In this case, it’s the \(-9\).
• Add 9 to both sides of the equation to move it away from the variable (\[3j - 9 + 9 = 12 + 9\]). This simplifies to \(3j = 21\).
• Next, divide both sides by 3, which is the coefficient of \(j\) (\[\frac{3j}{3} = \frac{21}{3}\]). This gives us \(j = 7\).

By using these steps, we have successfully isolated the variable \(j\), allowing us to find its value.

Checking Solutions

Once you've solved an equation, it's crucial to ensure your solution is correct. This process is known as checking solutions and involves substituting the found value back into the original equation.
Let's see how this works with our earlier example, where we found \(j = 7\). Start with the original equation: \(3j - 9 = 12\).

• Substitute \(j = 7\) into the equation: \(3(7) - 9 = 12\).
• Calculate the left side: this gives \(21 - 9 = 12\).
• The equation holds true since \(12 = 12\), confirming our solution is correct.

Checking ensures that the obtained solution satisfies the equation, providing confidence in the correctness of the result.

Mathematical Operations

Solving equations requires performing various mathematical operations to manipulate and simplify the expressions. Here are some key operations and their roles:

• **Addition and Subtraction**: Used to eliminate terms from one side of the equation. For instance, adding 9 helped remove the \(-9\) from \(3j - 9 = 12\).
• **Multiplication and Division**: Used to isolate the variable by canceling out coefficients. Dividing by 3 in the step \(3j = 21\) simplified to \(j = 7\).
• **Substitution**: This is not an operation per se in solving but crucial in checking solutions. Substituting the found solution back into the original equation ensures its validity.

Understanding these operations and when to apply them is key to solving equations effectively. Proper manipulation through these basic operations ensures that we can solve for unknown variables and verify our solutions with confidence.