Problem 53 - Fundamentals of Mathematics

Question

Solve the equation \(x-14+8=-2\).

Step-by-Step Solution

Answer

The solution is \( x = 4 \).

Step 1: Simplify the Equation

Start by simplifying the left side of the equation. Combine any like terms present. Here, you can simplify the expression by combining \( -14 + 8 \) to get \( -6 \), so the equation becomes: \( x - 6 = -2 \).

Step 2: Isolate the Variable

To solve for \( x \), you need to isolate \( x \) on one side of the equation. Add 6 to both sides of the equation to get rid of the -6 term on the left side. This gives you: \( x - 6 + 6 = -2 + 6 \), which simplifies to \( x = 4 \).

Step 3: Verify the Solution

To ensure that our solution is correct, substitute the value of \( x = 4 \) back into the original equation. The original equation is: \( x - 14 + 8 = -2 \). Substitute \( x=4 \) to get:\( 4 - 14 + 8 = -2 \).Calculate the left side: \( 4 - 14 + 8 = -2 \), which is true.

Key Concepts

Solving Equations

Equations are statements asserting two expressions are equal. Solving an equation involves finding the value of the variable that makes this statement true. In simpler terms, it is about discovering the 'unknown'. The equation we looked at was \( x - 14 + 8 = -2 \). The goal is to find the value of \( x \) that makes both sides of the equation equal.

To tackle this equation, you follow specific steps:

• Identify and simplify both sides of the equation if necessary.
• Use algebraic operations to simplify the equation further.
• Isolate the variable to find its value.

Remember that the operations you perform on one side of the equation, you must also perform on the other side. This keeps the equation balanced, maintaining the equality.

Simplifying Expressions

Simplifying makes expressions easier to work with by reducing them to their simplest form. In the equation \( x - 14 + 8 = -2 \), simplifying involved combining like terms, which simplifies understanding and solving further. Like terms are terms that have the same variables raised to the same power.Let’s see how simplification works:

• Observe the equation: \( x - 14 + 8 \). Here, \(-14\) and \(8\) are constants.
• Add \(-14\) and \(8\) together. This results in \(-6\), making the equation: \( x - 6 = -2 \).

This simplification step helps in making the equation more straightforward, preparing it for the next steps in solving.

Isolating Variables

Isolating a variable means manipulating the equation so that the variable is by itself on one side of the equation. The equation becomes much clearer when you isolate the variable, and you can easily see what value it holds.For our equation, \( x - 6 = -2 \), to isolate \( x \), you needed to:

• Add \(6\) to both sides of the equation. This action cancels out \(-6\) on the left side:
• This results in: \( x = -2 + 6 \).
• After performing the addition, you get \( x = 4 \).

Once the variable has been isolated, you have found the solution to the equation, which you can always verify by substituting back into the original equation to check its correctness.