Problem 8 - Elementary Algebra

Question

Is the given value a solution to the linear equation? $$ -13 x-4=-5 ; \quad x=-3 $$

Step-by-Step Solution

Answer

The value \( x = -3 \) is not a solution because it makes the left side of the equation 35, not -5.

Step 1: Understand the Problem

We need to determine if the given value \( x = -3 \) is a solution to the linear equation \( -13x - 4 = -5 \). This means we should substitute \( x = -3 \) into the equation and check if both sides are equal.

Step 2: Substitute the Value

Substitute \( x = -3 \) into the left side of the equation: \[-13(-3) - 4\]

Step 3: Simplify Substitution

Calculate the expression: \[-13(-3) = 39\] which changes the equation to \[39 - 4\]

Step 4: Further Simplify the Expression

Simplify \(39 - 4\):\[39 - 4 = 35\].This is the left side of the equation after substitution.

Step 5: Compare Both Sides

The right side of the original equation is \(-5\). Compare the left side \(35\) with \(-5\). Since \(35 eq -5\), \( x = -3 \) is not a solution.

Key Concepts

Solution of Equation

A solution to a linear equation is a value that makes the equation true when substituted for the variable. In simpler terms, it's like finding the 'key' that unlocks the equation. When you find the correct value and plug it into the equation, both sides should be equal. This balance of left and right sides confirms we've found a solution.

For example, consider the equation:

• \(-13x - 4 = -5\)

When we suspect that \(x = -3\) might be a solution, we test it by substituting \(-3\) for \(x\) and check if both sides are equal. In this case, substituting doesn't give us equal sides, indicating that \(x = -3\) is not a solution to the equation.

Substitution Method

The substitution method is a way to tackle equations by replacing a variable with a given number or expression. It's like filling in blanks in a sentence to see if it makes sense. This approach is especially handy for checking potential solutions.

Here's how it works with our equation example:

• The original equation is \(-13x - 4 = -5\).
• We substitute \(-3\) for \(x\) to see if it satisfies the equation.
• This means we replace every \(x\) with \(-3\) and calculate.

We get a new expression: \(-13(-3) - 4\). Solving this will either balance the equation (show it's a solution) or not (prove it's not a solution). It's a straightforward technique that gives results quickly.

Algebraic Manipulation

Algebraic manipulation involves working with mathematical symbols to simplify or solve equations. It's the toolset mathematicians use to break down and reconfigure equations. In our context, it's about transforming the equation step by step until it's easier to understand.

Here’s how it goes with our problem:

• We first compute \(-13(-3)\), which results in \(39\).
• Then, we perform subtraction: \(39 - 4\), leading to \(35\).

These steps manipulate the original equation in a way that simplifies the left side, allowing for an easy comparison with the right side. Comparing \(35\) with \(-5\) shows they differ significantly, indicating our initial assumption for \(x\) was incorrect. Through manipulation, equations gradually reveal whether proposed solutions fit.