Problem 48 - Algebra for College Students

Question

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction. \(4(x+2)-8 x-5=-3 x+9-2(x+6)\)

Step-by-Step Solution

Answer

The solution is x = -6. The equation is neither an identity nor a contradiction.

Step 1: Distribute

Distribute the constants through the parentheses on both sides of the equation. The equation becomes: 4(x + 2) - 8x - 5 = -3x + 9 - 2(x + 6) Which simplifies to: 4x + 8 - 8x - 5 = -3x + 9 - 2x - 12

Step 2: Combine Like Terms

Combine the like terms on both sides of the equation.On the left side, combine 4x and -8x, also combine 8 and -5:-4x + 3 = -5x -3

Step 3: Solve for x

Add 5x to both sides of the equation to get all x terms on one side:-4x + 5x + 3 = -5x + 5x - 3This simplifies to:x + 3 = -3 Subtract 3 from both sides to solve for x:x = -6

Step 4: Check the Solution

Substitute the value of x back into the original equation to verify it.Original equation: 4(x + 2) - 8x - 5 = -3x + 9 - 2(x + 6)Substitute x = -6:4(-6 + 2) - 8(-6) - 5 = -3(-6) + 9 - 2(-6 + 6)Simplify both sides:4(-4) + 48 - 5 = 18 + 9 - 0 -16 + 48 - 5 = 2727 = 27The solution is verified.

Key Concepts

Distributive Property

The distributive property is essential for solving linear equations. It allows you to remove parentheses by distributing a factor across the terms inside the parentheses. For the equation given in the exercise, the distributive property helps to simplify both sides.

Applying the distributive property to the left side, we get:

4(x + 2) - 8x - 5 = 4x + 8 - 8x - 5

On the right side, distribute -2 across (x + 6):

-3x + 9 - 2(x + 6) = -3x + 9 - 2x - 12

The distributive property transforms the equation into a simpler form, making it easier to combine like terms.

Combining Like Terms

Combining like terms is an important step when simplifying linear equations. Like terms are terms that have the same variable raised to the same power. In the example, after using the distributive property:

4x + 8 - 8x - 5 = -3x + 9 - 2x - 12

We then combine like terms on both sides of the equation. On the left side, combine 4x and -8x, and 8 and -5:

4x - 8x + 8 - 5 = -4x + 3

On the right side, combine -3x and -2x, and 9 and -12:

-3x - 2x + 9 - 12 = -5x - 3

The equation now simplifies to:

-4x + 3 = -5x - 3

Combining like terms makes the equation more manageable and brings you closer to solving for the variable.

Checking Solutions

Checking the solution is a crucial final step that ensures the accuracy of your answer. To verify the solution, substitute the value of x back into the original equation. If both sides of the equation are equal, your solution is correct. Let’s check the solution x = -6 for our equation:

Original equation:
4(x + 2) - 8x - 5 = -3x + 9 - 2(x + 6)

Substitute x = -6:

4(-6 + 2) - 8(-6) - 5 = -3(-6) + 9 - 2(-6 + 6)

Simplify both sides:

4(-4) + 48 - 5 = 18 + 9 - 0

-16 + 48 - 5 = 27
27 = 27

Since both sides are equal, the solution x = -6 is verified. Always check your solutions to avoid mistakes and ensure your answer is correct.