Problem 33 - Algebra for College Students

Question

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction. \(-2(x+3)=-6(x+7)\)

Step-by-Step Solution

Answer

The solution is negative 9. It is neither an identity nor a contradiction.

Step 1: Distribute

First, distribute the egative 2 on both sides of the equation. This means multiplying egative 2 with each term inside the parentheses: egative 2(x + 3) = egative 2x - 6 and egative 6(x + 7) = egative 6x - 42.

Step 2: Simplify Both Sides

After distributing, we have: egative 2x - 6 = egative 6x - 42.

Step 3: Add 6x to Both Sides

To isolate the variable, add 6x to both sides of the equation: egative 2x - 6 + 6x = egative 6x - 42 + 6x. Simplifying gives: 4x - 6 = -42.

Step 4: Add 6 to Both Sides

Next, add 6 to both sides to further isolate the variable: 4x - 6 + 6 = -42 + 6. Simplifying gives: 4x = -36.

Step 5: Divide by 4

Finally, divide both sides by 4 to solve for x: 4x / 4 = -36 / 4. Simplifying gives: x = -9.

Step 6: Check the Solution

To check the solution, substitute x = -9 back into the original equation: negative 2(-9 + 3) = negative 6(-9 + 7). Simplifying both sides we get: negative 2(-6) = negative 6(-2), which simplifies to 12 = 12. Therefore, the solution x = -9 is correct.

Step 7: Identify the Equation

Since the equation has a valid solution, it is neither an identity nor a contradiction.

Key Concepts

Distributive Property

The distributive property is essential when solving linear equations that involve parentheses. It states that multiplying a sum by a number is the same as multiplying each addend individually and then adding the results. In the context of the given problem, we use the distributive property to expand both sides of the equation \(-2(x+3)=-6(x+7)\) as follows:

• First, distribute \(-2\) across \(x+3\): \(-2(x + 3) = -2x - 6\)
• Next, distribute \(-6\) across \(x + 7\): \(-6(x + 7) = -6x - 42\)

This simplifies the equation to: \(-2x - 6 = -6x - 42\).By applying the distributive property, we simplify complex expressions and lay the groundwork for isolating the variable.

Isolating Variables

Isolating the variable is a fundamental step in solving linear equations. Our goal is to move all terms involving the variable to one side of the equation and constants to the other. Let’s see how this works in our exercise:

• Add \(6x\) to both sides: \(-2x - 6 + 6x = -6x - 42 + 6x\). This simplifies to \(4x - 6 = -42\).
• Next, add 6 to both sides to further isolate the variable: \(4x - 6 + 6 = -42 + 6\). This simplifies to \(4x = -36\).
• Finally, divide both sides by 4 to solve for \(x\): \(4x / 4 = -36 / 4\). Simplifying gives: \(x = -9\).

Isolating the variable is like peeling an onion, layer by layer, to reveal the solution beneath.

Checking Solutions

Once you have found a solution, it's crucial to check if it satisfies the original equation. This step verifies your work and ensures accuracy. In our exercise, we substitute \(x = -9\) back into the original equation to check our solution:

• Substitute into the left-hand side: \(-2(-9 + 3)\)
• Substitute into the right-hand side: \(-6(-9 + 7)\)
• Both simplify to: \(-2(-6) = 12\) and \(-6(-2) = 12\)

Since both sides equal \(12\), our solution \(x = -9\) is verified as correct. This step ensures that the solution we found actually solves the original equation.

Types of Equations

Different types of equations can yield different solutions. Here are a few to note:

• Unique Solution: Most linear equations will have a single, unique solution. In our example, \(x = -9\) is the unique solution.
• Identities: Some equations are true for all values of the variable, like \(2x + 3 = 2x + 3\). These are called identities and have infinite solutions.
• Contradictions: Some equations, such as \(x + 2 = x + 3\), are false for all values of the variable. These are contradictions and have no solutions.

In our exercise, since we obtained a valid solution, we determined that the equation is neither an identity nor a contradiction. Understanding these types helps in identifying the nature of the solutions we are working towards.