Problem 5 - Algebra A Combined Function

Question

Solve each equation. Check each solution. $$ -11=3+x $$

Step-by-Step Solution

Answer

The solution is \(x = -14\). Substitute back to verify correctness.

Step 1: Isolate the Variable

The goal is to solve for the variable \(x\). Start by isolating \(x\) on one side of the equation. The equation is given as \(-11 = 3 + x\). To isolate \(x\), subtract \(3\) from both sides:\[-11 - 3 = x\]

Step 2: Simplify the Equation

Now simplify the left side of the equation:\[-11 - 3 = -14\]So, \(x = -14\).

Step 3: Verify the Solution

Substitute \(x = -14\) back into the original equation to verify:\[-11 = 3 + (-14)\]Simplify the right side:\[3 - 14 = -11\]Since both sides are equal, \(x = -14\) is the correct solution.

Key Concepts

Variable Isolation

In solving linear equations, one of the most fundamental steps is isolation of the variable. The variable is generally the unknown element in an equation that you are solving for. Here, it is important to arrange the equation so that the variable ends up alone on one side. Let's look at the example equation: - \(-11 = 3 + x\).To isolate the variable \(x\), you'll want to eliminate surrounding numbers or variables that are adding to or subtracting from \(x\). In this particular problem, \(x\) is on the right side, together with \(3\). The operation opposite to addition is subtraction, so we subtract \(3\) from both sides:- \(-11 - 3 = x\).Subtracting \(3\) from both sides allows \(x\) to stand alone, and thus be isolated. Always remember, your goal is to find \(x\), so manipulating the equation to get \(x\) alone is key.

Equation Simplification

Once you have isolated the variable, the next step is to simplify the equation. This involves performing the arithmetic operations on both sides of the equation to simplify it as much as possible. In the step-by-step solution, we are looking at the equation derived from the previous step:- \(-11 - 3 = x\).Simplifying means you calculate the expression on the left:- \(-11 - 3 = -14\).Once simplified, we find that \(x\) equals \(-14\). This step ensures that the equation is as simple as possible, containing only the variable \(x\) on one side and a simple number on the other.

Solution Verification

After finding a potential solution, there's an essential step: verifying your solution. Verifying means checking if the value of \(x\) you found satisfies the original equation. To do this, substitute the value back into the original equation:- Start with \(-11 = 3 + x\).- Substitute \(x = -14\), so it becomes \(-11 = 3 + (-14)\).When you simplify the right side:- \(3 - 14 = -11\).Since both the left and right sides are equal, it confirms that \(x = -14\) is indeed the correct solution. By verifying your solution, you ensure that there are no mistakes and that your answer is accurate. This step is crucial for increasing your confidence in solving linear equations. Remember, always checking your solution helps prevent errors and builds a solid understanding of the problem at hand.