Problem 10 - Prealgebra

Question

Solve each equation. Check your solution. $$-6 j+4+3 j=-23$$

Step-by-Step Solution

Answer

The solution is \(j = 9\).

Step 1: Combine Like Terms

First, identify and combine the like terms on the left side of the equation. The terms are \(-6j\) and \(+3j\). When you combine these, the equation becomes: \[-3j + 4 = -23\]

Step 2: Isolate the Variable Term

To isolate the variable term, you need to get rid of the constant on the left side. Subtract 4 from both sides of the equation: \[-3j + 4 - 4 = -23 - 4\] This simplifies to: \[-3j = -27\]

Step 3: Solve for the Variable

Now, solve for \(j\) by dividing both sides of the equation by -3 to get:\[j = \frac{-27}{-3}\]This simplifies to:\[j = 9\]

Step 4: Check the Solution

Substitute \(j = 9\) back into the original equation to verify the solution:\[-6(9) + 4 + 3(9) = -23\]Calculate step by step: \[ -54 + 4 + 27 = -23\]\[-54 + 31 = -23\]\[-23 = -23\]The left side equals the right side, so \(j = 9\) is correct.

Key Concepts

Solving Equations

Solving equations is an essential prealgebra skill that helps in finding the value of the unknown variable. In our example, we are solving the equation: \[-6j + 4 + 3j = -23\] The goal is to find the value of \(j\), which makes the equation true.
To solve, we follow a systematic approach:

• Perform operations to simplify both sides of the equation.
• Get all the terms with the variable on one side and constants on the other side.
• Isolate the variable by performing inverse operations.

By following these steps, we make the equation simpler and find the solution step by step.

Combining Like Terms

Combining like terms is key in simplifying equations. Like terms have the same variable raised to the same power. Here, the terms \(-6j\) and \(3j\) are like terms because they both contain the variable \(j\).
To combine like terms, you add or subtract their coefficients. For our problem:
- \(-6j\) and \(3j\) combine to give \(-3j\), since \(-6 + 3 = -3\).
This step reduces complexity and brings us closer to solving the equation. Once like terms are combined, we can proceed to isolate the variable term, making the equation simpler to solve.

Check Your Solution

Once you think you have a solution, it's crucial to check your work. Checking verifies that the found value truly satisfies the original equation. Here's how you do it:

• Substitute the solution back into the original equation.
• Perform the arithmetic to ensure both sides of the equation are equal.

For our example, substituting \(j = 9\) back:\[-6(9) + 4 + 3(9) = -23\] simplifies to \(-54 + 4 + 27 = -23\), or \(-23 = -23\).
Seeing the original equation hold true confirms that \(j = 9\) is the correct solution. This step ensures credibility, showing that the solution is logical and accurate.