Problem 37 - Algebra for College Students

Question

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction. \(2 x+3(x-4)=2(x-3)\)

Step-by-Step Solution

Answer

x = 2. The solution is valid as verified. It is neither an identity nor a contradiction.

Step 1: Distribute the terms

Distribute the terms inside the parentheses. Rewrite the equation: 2x + 3(x - 4) = 2(x - 3) This gives: 2x + 3x - 12 = 2x - 6

Step 2: Combine like terms

Combine the terms with the same variable on the left side of the equation. This results in: 5x - 12 = 2x - 6

Step 3: Move variables to one side

Subtract 2x from both sides to move variables to one side. 5x - 2x - 12 = -6 This simplifies to: 3x - 12 = -6

Step 4: Isolate the variable

Add 12 to both sides to isolate the variable term. 3x - 12 + 12 = -6 + 12 Simplifies to: 3x = 6

Step 5: Solve for x

Divide both sides by 3 to solve for x. 3x / 3 = 6 / 3 Thus, x = 2

Step 6: Check the solution

Substitute x = 2 back into the original equation to verify it. 2(2) + 3(2 - 4) = 2(2 - 3) Which simplifies to 4 + 3(-2) = 2(-1) This simplifies to 4 - 6 = -2 and to -2 = -2 Which is true, confirming the solution.

Step 7: Determine if identity or contradiction

Since the equation has a single solution x = 2, it is neither an identity nor a contradiction.

Key Concepts

Distributive Property

The distributive property is a powerful tool in algebra. It allows you to simplify equations by distributing a common factor across terms inside parentheses. In our exercise, we start with the equation:

2x + 3(x - 4) = 2(x - 3)

To apply the distributive property:

• Multiply 3 by each term inside the parentheses: 3 * x and 3 * -4

This gives:

2x + 3x - 12 = 2(x - 3)

Now, distribute 2 on the right side:

• Multiply 2 by each term inside the parentheses: 2 * x and 2 * -3

Which simplifies to:

2x + 3x - 12 = 2x - 6

Using the distributive property helps in breaking down complex expressions and making the equations easier to solve.

Combining Like Terms

Combining like terms simplifies an equation by merging terms with the same variables. This step makes the equation more manageable. In our example, after using the distributive property, we have:

2x + 3x - 12 = 2x - 6

On the left side, we combine the terms with x:

• 2x + 3x = 5x

This results in:

5x - 12 = 2x - 6

Combining like terms keeps equations tidy and helps visualize the necessary steps to isolate the variable. It's an essential skill for simplifying complex equations and solving them efficiently.

Isolating the Variable

Isolating the variable is the process of getting the unknown variable on one side of the equation and everything else on the other. This is a crucial step in solving equations. Continuing from our simplified equation:

5x - 12 = 2x - 6

We move the variable terms to one side by subtracting 2x from both sides:

• 5x - 2x = 3x
• 3x - 12 = -6

Next, we isolate the variable term by adding 12 to both sides:

3x - 12 + 12 = -6 + 12

We get:

3x = 6

Finally, solve for x by dividing both sides by 3:

• 3x / 3 = 6 / 3
• x = 2

Isolating the variable makes it easier to find the solution of an equation. This step turns a complex problem into a simple one, revealing the value of the unknown variable.