Problem 117 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ 3 x+2 x=6 $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{6}{5}\).

Step 1: Simplify the equation

Start by simplifying the left-hand side of the equation. Combine like terms by adding the coefficients of the same variable. In this case, both terms on the left side involve the variable \(x\). Add \(3x\) and \(2x\) to get \(5x\). Thus, the equation becomes \(5x = 6\).

Step 2: Isolate the variable

Now, isolate \(x\) by dividing both sides of the equation by the coefficient of \(x\), which is 5. This gives: \(x = \frac{6}{5}\).

Step 3: Verify the solution

Substitute \(x = \frac{6}{5}\) back into the original equation to ensure it holds true. \(3 \left(\frac{6}{5}\right) + 2 \left(\frac{6}{5}\right) = 6\). Simplifying each term, \(\frac{18}{5} + \frac{12}{5} = \frac{30}{5} = 6\), confirming that \(x = \frac{6}{5}\) is correct.

Key Concepts

Combining Like Terms

Combining like terms is an essential concept in solving equations because it helps to simplify the problem. When you combine like terms, you add or subtract terms that have the same variable raised to the same power. In the equation given in the exercise, both terms on the left-hand side involve the variable \(x\). Therefore, they are considered like terms.

• First, identify the terms that can be combined. In our example, these are \(3x\) and \(2x\).
• Next, add the coefficients of these terms. Coefficients are the numbers in front of the variables. Here, we add 3 and 2 to get 5.
• Finally, rewrite the expression using the combined terms. So, \(3x + 2x\) becomes \(5x\).

Simplifying the expression by combining like terms makes it easier to isolate the variable in the next steps of the solving process.

Isolating Variables

Isolating the variable is a crucial step to solve an equation. The goal here is to get the variable itself on one side of the equation, with everything else on the opposite side. This allows us to find the value of the variable. In the equation \(5x = 6\), the variable is already on one side, but it is still multiplied by a coefficient.

• To isolate \(x\), you need to get rid of the 5. Since the 5 is multiplying \(x\), divide both sides of the equation by 5.
• This division cancels out the coefficient on the left side, leaving \(x\) by itself: \(x = \frac{6}{5}\).

By isolating the variable, you can clearly see what \(x\) is equal to, solving the equation in a straightforward way.

Verifying Solutions

Once you've solved for the variable, it's important to verify that your solution is correct. Verifying solutions means plugging the found value back into the original equation to see if it satisfies the equation.

• First, substitute \(x = \frac{6}{5}\) back into the original equation \(3x + 2x = 6\).
• Calculate each term: \(3 \left(\frac{6}{5}\right)\) and \(2 \left(\frac{6}{5}\right)\).
• Adding them gives \(\frac{18}{5} + \frac{12}{5} = \frac{30}{5} = 6\).

The left side of the equation equals the right side, so your solution is verified. This step ensures that no errors have been made in the calculations, confirming the accuracy of your solution.