Problem 57 - Algebra 2

Question

Solve each equation or inequality. Check your solutions. $$ 2 x+7=-3 $$

Step-by-Step Solution

Answer

The solution is \(x = -5\).

Step 1: Move Constant to Other Side

Given the equation \(2x + 7 = -3\), we first need to isolate the term with \(x\). Subtract 7 from both sides to get rid of the constant on the left.\[2x + 7 - 7 = -3 - 7\]This simplifies to:\[2x = -10\]

Step 2: Isolate the Variable

To solve for \(x\), divide both sides of the equation by 2.\[\frac{2x}{2} = \frac{-10}{2}\]This simplifies to:\[x = -5\]

Step 3: Verify the Solution

Substitute \(x = -5\) back into the original equation to check if it satisfies the equation.\[2(-5) + 7 = -3\]Calculate:\[-10 + 7 = -3\]This equates to:\[-3 = -3\]Since both sides of the equation are equal, \(x = -5\) is the correct solution.

Key Concepts

Isolating Variables

When solving linear equations, the process often begins with isolating the variable on one side of the equation. The main goal is to have the variable term stand alone with a coefficient of 1 if possible. By rearranging the equation, you can see the numerical value of the variable. For instance, in the equation \(2x + 7 = -3\), the first step to isolate \(x\) is by getting rid of the constant term 7. You can do this by subtracting 7 from both sides of the equation. This gives you:

• \(2x + 7 - 7 = -3 - 7\)
• This simplifies to \(2x = -10\)

Next, you have \(2x\) isolated but not \(x\). To finally isolate \(x\), divide both sides by 2:

• \(\frac{2x}{2} = \frac{-10}{2}\)
• This results in \(x = -5\)

Once the variable is isolated and its value found, you are ready to check if the solution is correct.

Checking Solutions

Checking your solution is an essential part of solving equations. This step helps ensure that the solution you've calculated is indeed correct. To check the solution, substitute the calculated value back into the original equation. Substituting \(x = -5\) into \(2x + 7 = -3\) gives:

• \(2(-5) + 7 = -3\)

Now calculate the left side of this substitution:

• \(-10 + 7 = -3\)
• This simplifies to \(-3 = -3\)

Since both sides of the equation are equal, your solution of \(x = -5\) is confirmed as correct. Performing this substitution ensures the reliability of your solution, verifying any mistakes or misconceptions you may have had during calculation.

Substitution in Equations

Substitution is a powerful technique used not only to check solutions but also useful in solving systems of equations or applying initial conditions. In the process of checking solutions, you replace the variable with its found value in the equation, as demonstrated earlier. This method is straightforward in linear equations:

• Insert the value back (e.g., \(x = -5\)) in place of \(x\)
• Perform the arithmetic operations and simplify
• Ensure both sides are equal to confirm the solution

Substitution is beneficial because it allows you to verify the accuracy of your solution. By practicing substitution, you will gain confidence in your ability to reach the correct answers in different mathematical contexts.