Problem 15 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. $$10 x+10=0$$

Step-by-Step Solution

Answer

The solution to the equation \(10x + 10 = 0\) is \(x = -1\).

Step 1: Isolate the variable term

In order to isolate the variable \(x\), first step is to exclude the constant from the equation. Given the equation \(10x + 10 = 0\), the term without the variable is subtracted from both sides, which would leave us with \(10x = -10\).

Step 2: Solve for the variable

Now that we have \(10x = -10\), we can solve for the variable \(x\) by dividing both sides by 10. This gives \(x = -10/10 = -1\).

Step 3: Check your solution

To make sure the solution is correct, substitute \(x = -1\) back into the original equation. Substituting we get \(10*(-1) + 10 = -10 + 10 = 0\), which is the right hand side of the original equation, hence \(-1\) is indeed a solution to the equation.

Key Concepts

Isolating the Variable

When we talk about isolating the variable in a linear equation, such as the one given, the goal is to have the variable all by itself on one side of the equation. This makes it straightforward to determine its value. In our example, the equation is:

• \(10x + 10 = 0\)

To isolate the variable \(x\), we first need to remove the constant term, which is 10, from the same side as \(x\). This is done by performing the same arithmetic operation on both sides of the equation. In this case, we subtract 10 from both sides:

• \(10x + 10 - 10 = 0 - 10\)

This simplifies to:

• \(10x = -10\)

Now that \(x\) is isolated with its coefficient (the number in front of \(x\)), we can proceed to solve for \(x\). This step is crucial because it sets the stage for finding the value of the variable and completing the solution.

Substitution Method

Once we have isolated the variable, we solve for it by using one more arithmetic operation. In our case, we have:

• \(10x = -10\)

To find the value of \(x\), we divide both sides of the equation by 10, which is the coefficient of \(x\):

• \(x = \frac{-10}{10}\)

This simplifies to:

• \(x = -1\)

Having found \(x = -1\), it's important to verify this solution. Substitution involves taking this value and putting it back into the original equation to ensure it satisfies the equation. This confirms the correctness of our calculations.

Checking Solutions

After finding a solution, such as \(x = -1\), checking your work ensures accuracy. This involves substituting the calculated value back into the original equation to see if both sides are equal. For our problem, the check would look like this:

• Substitute \(x = -1\) into \(10x + 10 = 0\)
• The equation becomes \(10(-1) + 10 = -10 + 10\)
• This simplifies to \(0 = 0\)

Both sides of the equation are equal, confirming that \(-1\) is indeed the correct and valid solution. Need for checking solutions cannot be overstated, as it provides assurance that your solution process is correct. It helps to catch any small errors that might have occurred during calculations and reinforces understanding of the problem-solving process.