Problem 40 - Elementary Algebra

Question

Solve each of the conditional equations. $$ x+17=-426 $$

Step-by-Step Solution

Answer

Answer: The value of x that satisfies the given equation is -443.

Step 1: Identify the given equation

The given equation is: $$ x + 17 = -426 $$

Step 2: Isolate x on one side of the equation

In order to isolate x on one side of the equation, subtract 17 from both sides: $$ x + 17 - 17 = -426 - 17 $$

Step 3: Simplify the equation

Now, simplify the equation by performing the subtraction: $$ x = -443 $$

Step 4: Check the solution

To verify that our solution is correct, plug x = -443 back into the original equation: $$ (-443) + 17 = -426 $$ Since the equation holds true, our solution x = -443 is correct.

Key Concepts

Isolation of Variables

In solving equations, one fundamental step is isolating the variable. The goal here is to have the variable on one side of the equation and everything else on the opposite side. This process helps you focus on what value the variable must have to make the equation true. Consider the equation given:
\[ x + 17 = -426 \]
To isolate \( x \), you need to remove any number alongside it. This often involves performing the opposite mathematical operation.

• Addition's opposite is subtraction.
• If there's division, you use multiplication.
• For multiplication, apply division.

In this case, since \( 17 \) is added to \( x \), we subtract \( 17 \) from both sides of the equation:
\[ x + 17 - 17 = -426 - 17 \].
This subtraction simplifies the effort to simply breathing space for \( x \) alone. Now, \( x \) is on one side by itself:
\[ x = -443 \].

Simplification

Once the variable is isolated, the next step is simplification. Simplification involves breaking down expressions to make them more straightforward and concise, often focused on handling any arithmetic left over.

• After the subtraction, we have the equation in a simpler form.
• Perform arithmetic as needed, sometimes combining like terms or resolving basic sums. In this exercise:

The operation on both sides was subtraction, leading to \(-426 - 17\). Calculating this results in \(-443\):
\[ x = -443 \].
Simplification is crucial as it ensures you don't miscalculate steps and enables a clear view of what needs resolving.

Verification of Solutions

Verification serves as a checkpoint to ensure that the obtained solution truly satisfies the original equation. This involves plugging your supposed solution back into the initial equation to check if it makes the equation true. For this example, after finding \( x = -443 \), substitute \( x \) back:
\[ (-443) + 17 \].
The result should equal the other side of the equation, \(-426\). If:
\[ (-443) + 17 = -426 \],
the solution is verified as correct. Verification is a critical step as it confirms your solution’s accuracy and completeness. It reinforces confidence in each step taken during solving, ensuring no arithmetic or logic errors were made.