Problem 12 - Elementary Algebra

Question

Is the given value a solution to the linear equation? $$ 3(2 x+1)=-4 x-3 ; x=-12 $$

Step-by-Step Solution

Answer

No, \( x = -12 \) is not a solution to the equation.

Step 1: Substitute the Given Value into the Equation

Substitute the given value of \( x = -12 \) into the original equation:\[ 3(2(-12) + 1) = -4(-12) - 3 \]

Step 2: Simplify Both Sides of the Equation

Simplify the left side of the equation:- Start with the expression inside the parentheses: \( 2(-12) + 1 = -24 + 1 = -23 \)- Multiply by 3: \( 3(-23) = -69 \)The left side simplifies to \( -69 \).Now simplify the right side of the equation:- Multiply -4 times -12: \( -4(-12) = 48 \)- Subtract 3: \( 48 - 3 = 45 \)The right side simplifies to \( 45 \).

Step 3: Compare Both Sides of the Equation

Compare the simplified left side and right side:The left side is \( -69 \) and the right side is \( 45 \).Since \( -69 eq 45 \), the two sides are not equal.

Key Concepts

Solution Checking

Checking if a given value satisfies a linear equation is an important skill in mathematics. Here, the process involves substituting the given value into the equation and simplifying both sides to see if they match. In our example, we need to determine whether \( x = -12 \) is a solution to the equation \( 3(2x + 1) = -4x - 3 \).
Start by replacing \( x \) with \( -12 \) in the equation. Then, simplify the mathematical expressions on both sides.

• If both sides are equal after simplification, the given value is a solution to the equation.
• If they are not equal, the given value is not a solution.

In this exercise, substituting \( x = -12 \) into the equation results in \( -69 \) on the left side and \( 45 \) on the right side. Since these are not equal, \( x = -12 \) is not a solution.
It’s a useful method to ensure your calculations are correct and can also help identify mistakes if something doesn’t add up.

Equation Simplification

Simplifying equations is a key part of solving them and helps in understanding if a value is truly a solution. This process entails performing all possible arithmetic to condense the equation into its simplest form.
In our exercise, the original equation \( 3(2x + 1) = -4x - 3 \) has been simplified to make solving easier. Let's break it down:

• Left Side: We start with the expression \( 2x + 1 \) within the parentheses.
• Substitute \( x \) with \( -12 \): \( 2(-12) + 1 = -24 + 1 = -23 \).
• Multiply by 3: \( 3(-23) = -69 \).
• Right Side: First, multiply \( -4 \) by \( -12 \): \( -4(-12) = 48 \).
• Then, subtract 3: \( 48 - 3 = 45 \).

Through simplification, each side of the equation is reduced to a single number. This makes it easy to directly compare both sides to check for equality, confirming if the provided solution satisfies the equation.

Substitution Method

The substitution method is a straightforward algebraic technique often used in solving linear equations. It involves replacing a variable with a given number. This makes it possible to turn an equation with a variable into a simpler arithmetic problem.

• In this exercise, substitution was the first step: replacing \( x \) with \( -12 \).
• This changes the equation from \( 3(2x + 1) = -4x - 3 \) into \( 3(2(-12) + 1) = -4(-12) - 3 \).
• This allows you to handle easier math operations.

By following this method, it's easier to see exactly how a given value affects the overall equation. Just plug in the value and simplify to find out the result. If the simplified sides end up equal, the substitution provides a solution, confirming the given number works in the equation. Otherwise, it shows the provided value doesn't hold up, allowing for troubleshooting and further exploration.