Problem 69 - Algebra A Combined Function

Question

Decide whether the given number is a solution of the given equation. \(x-9=5 ;-4\)

Step-by-Step Solution

Answer

-4 is not a solution of the equation.

Step 1: Understand the Problem

We need to determine if the given number, which is -4, satisfies the equation \(x - 9 = 5\). This means substituting -4 in place of \(x\) and checking if the equation holds true.

Step 2: Substitute the Given Number

Substitute \(x = -4\) into the equation. The equation becomes \(-4 - 9 = 5\).

Step 3: Simplify the Left Side

Simplify \(-4 - 9\) on the left side of the equation. This equals \(-13\).

Step 4: Compare Both Sides

After simplifying, we have \(-13 = 5\). Since \(-13\) is not equal to \(5\), this tells us that substituting \(x = -4\) does not satisfy the equation.

Key Concepts

Substitution Method

The substitution method is a powerful tool in solving equations that involves replacing a variable with a given number to check if it satisfies the equation. Imagine you have an equation with an unknown value, like a puzzle. The objective is to see if the specific piece you have, here -4, fits the puzzle perfectly. To apply this method:

• Take the equation you need to solve, which in our case is \(x - 9 = 5\).
• Replace \(x\) with the number you are testing, \(-4\).
• Substitute it directly into the equation, transforming it into \(-4 - 9 = 5\).

Now, we just need to check if our puzzle piece fits. If the equation remains true, then -4 is a solution. The substitution method helps quickly determine the validity of potential solutions by direct replacement.
Use this method anytime you need to verify solutions for simple problems.

Simplification

Simplification refers to the process of reducing a complex expression or equation into a simpler form. This is like cleaning up a messy room to easily find what's needed. In our equation \(-4 - 9 = 5\), simplifying helps us see more clearly if each side is equal.Here's how you simplify:

• Look at the left side of the equation, \(-4 - 9\).
• Combine like terms, simplifying it to \(-13\).

This step ensures the equation is less cluttered, exposing whether the left-side value equals the right-side value.
After simplification, if both sides of the equation are equal, the solution is valid. In our case, simplifying showed that \(-13\) does not equal \(5\). This means \(-4\) is not the correct solution.Simplification is crucial in many mathematical processes to clarify complex expressions.

Linear Equations

A linear equation is essentially an equation that forms a straight line when graphed. These equations typically appear in the format of \(ax + b = c\), where both \(a\) and \(b\) are constants and \(x\) is the variable to solve for. They always involve the first power of the variable.For our example, the linear equation \(x - 9 = 5\) fits perfectly into this definition.

• Linear equations always produce straight lines when plotted on a graph.
• These equations are straightforward, often solving by simple operations like addition, subtraction, multiplication, or division.

Understanding linear equations is fundamental because they lay the groundwork for more complicated equations you'll encounter later.
Solving them requires basic operations, often involving checking or finding the missing variable value that satisfies the equation.Once you master linear equations, tackling more complex algebraic problems will become much easier.