Problem 5 - Elementary Algebra

Question

For Problems \(1-72\), use the properties of equality to help solve each equation. $$ -7=x+2 $$

Step-by-Step Solution

Answer

The value of \(x\) is \(-9\).

Step 1: Identify the Equation

The given equation is \(-7 = x + 2\). We need to solve for \(x\).

Step 2: Isolate the Variable

To isolate \(x\), subtract \(2\) from both sides of the equation. This will help us determine the value of \(x\).\(-7 - 2 = x + 2 - 2\)

Step 3: Simplify Both Sides

Simplify the expression on both sides of the equation after subtracting \(2\):\(-9 = x\)

Step 4: Verify the Solution

Plug \(x = -9\) back into the original equation to check if it makes the equation true.Substitute: \(-7 = (-9) + 2\)\(-7 = -7\). This verifies our solution is correct.

Key Concepts

Solving Equations

When you are solving equations, your goal is to find the value of the variable that makes the equation true. Equations are mathematical statements indicating that two expressions are equal to each other, often presented as something like this: \[ a = b \].The main principle of solving equations is to maintain balance. Just like a seesaw, both sides have to be equal for the equation to hold true.To solve an equation:

• Identify the operations being applied to the variable.
• Use inverse operations to "undo" these actions and solve for the variable.
• Utilize properties of equality to balance both sides throughout these changes.

For example, in the equation \(-7 = x + 2\), the variable, \(x\), has an added \(2\). Solving involves getting \(x\) on its own by removing \(2\) through subtraction.

Isolate the Variable

Isolation of the variable is a key step in solving equations. The idea is to rearrange the equation so that the variable is by itself on one side. This allows you to clearly see the variable's value. To isolate the variable:

• Identify any numbers or terms attached to the variable.
• Apply the inverse operation to both sides of the equation.

In the given equation, \(-7 = x + 2\), notice that \(2\) is added to \(x\). To isolate \(x\), you need to do the opposite of adding \(2\), which is subtracting \(2\) from both sides. This subtraction transforms the equation as follows:\[ -7 - 2 = x + 2 - 2 \]Simplifying this results in \(-9 = x\), which tells you that the value of \(x\) is \(-9\). Remember, always perform the same operation on both sides to keep the equation balanced.

Verify the Solution

Once you have isolated the variable and found a solution, the next step is to verify it. Verifying the solution ensures that this value actually works in the original equation. It's like double-checking your math homework just to make sure everything adds up.Here's how you verify it:

• Take the value you've found for the variable and substitute it back into the original equation.
• Calculate both sides of the equation to check if they are equal.

In our example, we found \(x = -9\). Substitute \(-9\) back into the original equation \(-7 = x + 2\):\[ -7 = (-9) + 2 \]This simplifies to \(-7 = -7\), confirming that our found solution is indeed correct. Verifying solutions not only gives you confidence in your answer but also reinforces your understanding of algebraic principles. Always verify to ensure accuracy!