Problem 6 - Algebra 1: Concepts and Skills

Question

Solve the equation. Check your solution in the original equation. $$ 3 x=18 $$

Step-by-Step Solution

Answer

The solution is \(x = 6\).

Step 1: Dividing the Equation

The first step towards solving the given equation is to isolate the variable, x. For that, divide both sides of the equation by 3: \(x = 18/3\).

Step 2: Solving for x

Now, perform the division operation on the right side of the equation to get the value of x: \(x = 6\).

Step 3: Checking the Solution

To check if the solution is correct, substitute \(x = 6\) back into the original equation: \(3(6) = 18\). The equation holds true, indicating that the solution is correct.

Key Concepts

Isolate the Variable

When it comes to solving linear equations, one of the fundamental steps is to 'isolate the variable'. This means you need to get the variable you are solving for by itself on one side of the equation, effectively separating it from the other numbers or variables in the equation. Take for instance the equation from the exercise, 3x = 18. Here, the goal is to determine the value of 'x'. To isolate 'x', you must perform operations that will undo the operations being applied to 'x'.

In this equation, 'x' is being multiplied by 3. To isolate 'x', you need to perform the opposite operation, which in this case is division. This is because division is the inverse operation of multiplication. By dividing both sides of the equation by 3, you neutralize the multiplication of 'x' by 3, leaving 'x' by itself: \[ x = \frac{18}{3} \]. This practice of 'undoing' the operation to isolate the variable is the heartbeat of solving linear equations.

Division Operation

After isolating the variable, the 'division operation' often comes into play, especially when the variable is multiplied by a number. Division is one of the four basic operations in arithmetic and is the process of finding how many times one number is contained within another.

Using our exercise as an example, once we've isolated 'x' to one side, we need to perform division to find its value: \[ x = \frac{18}{3} \]. In other words, we are asking, 'how many times does 3 go into 18?'. The answer to this division operation is 6. Understanding division is essential because it enables us to evenly distribute a number, thereby simplifying an equation to find the value of an unknown variable. Keep in mind that you can check if you've divided correctly by performing the inverse operation - multiplication - with your answer.

Check the Solution

Finally, it's crucial not to assume our solution is correct without verifying it. We 'check the solution' to ensure the value we obtained for the variable satisfies the original equation. To do this, we substitute our found value back into the original equation and perform the necessary operations to see if both sides of the equation remain equal.

In our example, if we substitute 'x' with 6 into the original equation, 3x = 18, we get \[ 3(6) = 18 \]. Simplifying the left side by multiplying 3 by 6 indeed gives us 18, which confirms that our solution is correct because it matches the right side of the equation. This check is a critical step as it validates our entire process, giving us confidence that our solution is accurate.