Problem 47 - Algebra for College Students

Question

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction. \(-3 x+6-5(x-1)=-5 x-(2 x-4)+5\)

Step-by-Step Solution

Answer

x = 2; the equation has a specific solution.

Step 1 - Distribute on both sides

First, distribute the constants through the parentheses on both sides of the equation.Left side: -3x + 6 - 5(x - 1) = -3x + 6 - 5x + 5 Right side: -5x - (2x - 4) + 5 = -5x - 2x + 4 + 5

Step 2 - Simplify both sides

Now, combine like terms on both sides of the equation.Left side: -3x - 5x + 6 + 5 = -8x + 11 Right side: -5x - 2x + 4 + 5 = -7x + 9

Step 3 - Move variable terms to one side

Add 8x to both sides of the equation to move all variable terms to the right side. -8x + 11 + 8x = -7x + 9 + 8x This simplifies to: 11 = x + 9

Step 4 - Solve for x

Subtract 9 from both sides of the equation to solve for x. 11 - 9 = x + 9 - 9 This gives: x = 2

Step 5 - Verify the solution

Substitute x = 2 into the original equation to check if both sides are equal. Original equation: -3x + 6 - 5(x - 1) = -5x - (2x - 4) + 5 Substituting x = 2: -3(2) + 6 - 5(2 - 1) = -5(2) - [2(2) - 4] + 5 Simplifies to: -6 + 6 - 5(1) = -10 - (4 - 4) + 5 Which further simplifies to: -5 = -5, confirming the solution is correct.

Step 6 - Identify the type of equation

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

Key Concepts

Distributive Property

The Distributive Property is crucial when dealing with linear equations that involve parentheses. This property states that you multiply each term inside the parentheses by the term outside it.
In our example, we start with -3x + 6 - 5(x - 1) = -5x - (2x - 4) + 5.
Let's apply the Distributive Property to both sides:

• Left side: -3x + 6 - 5(x - 1) transforms to -3x + 6 - 5x + 5
• Right side: -5x - (2x - 4) + 5 becomes -5x - 2x + 4 + 5

After distributing, the equation looks different, but it still holds the same value. Understanding this step is fundamental before moving on to simplify further.

Combining Like Terms

After distributing, the next step is to combine like terms. Like terms have the same variable raised to the same power. In our example, we have: Left side: -3x + 6 - 5x + 5
Right side: -5x - 2x + 4 + 5.
To combine them, group the x-terms together and the constant terms together:

• Left side: -3x - 5x + 6 + 5 simplifies to -8x + 11
• Right side: -5x - 2x + 4 + 5 simplifies to -7x + 9

This simplification is essential to make the equation easier to solve. It's simply a matter of arithmetic: adding or subtracting the coefficients of like terms.

Moving Variable Terms

After combining like terms, we need to move all variable terms to one side of the equation to solve for x. This often involves adding or subtracting terms on both sides of the equation.
In our example, -8x + 11 = -7x + 9.
To isolate x, add 8x to both sides:
-8x + 11 + 8x = -7x + 9 + 8x,
which simplifies to 11 = x + 9.
Now x is by itself on one side, making the equation ready to solve. This step is about moving terms strategically to isolate the variable, and it’s crucial for finding the solution.

Verifying Solutions

After solving for the variable, it’s important to verify the solution to ensure it's correct. This involves substituting the solution back into the original equation.
In our example, we found x = 2.
Let’s substitute x = 2 back into the original equation:
-3(2) + 6 - 5(2 - 1) = -5(2) - (2(2) - 4) + 5.
Simplify each part:

• Left side: -6 + 6 - 5(1) = -5
• Right side: -10 - (4 - 4) + 5 = -5

Both sides equal -5, confirming our solution x = 2 is correct. This verification step is essential to ensure no mistakes were made during the calculation. It validates that the solution satisfies the original equation.