Problem 27 - Algebra A Combined Function

Question

Solve each equation. Don't forget to first simplify each side of the equation, if possible. Check each solution. See Examples 5 through 7 . $$ -3(x-4)=-4 x $$

Step-by-Step Solution

Answer

The solution to the equation is: \(x = -12\).

Step 1: Distribute the Negative Sign

Begin by distributing the \[-3\] over \[(x-4)\] on the left side of the equation. This means multiplying \[-3\] with both terms inside the parenthesis:\[-3(x-4) = -3 \cdot x + (-3) \cdot (-4)\]This simplifies to:\[-3x + 12\]so the equation now becomes:\[-3x + 12 = -4x\]

Step 2: Rearrange Like Terms

Move all terms involving \(x\) to one side and constant terms to the other side to simplify the equation. Let's move \(-3x\) to the right side by adding \(3x\) to both sides:\[12 = -4x + 3x\]Simplify the right side:\[12 = -x\]

Step 3: Solve for x

To solve for \(x\), divide both sides by \(-1\):\[\frac{12}{-1} = x\]This yields:\[x = -12\]

Step 4: Verify the Solution

Substitute the calculated value \(x = -12\) back into the original equation to check if it holds true:\[-3(-12-4) = -4(-12)\]Simplify both sides:\[-3(-16) = 48\]\[48 = 48\]Since both sides of the equation are equal, the solution is verified as correct.

Key Concepts

Distributive Property

The distributive property is a foundational concept in algebra that allows us to break down expressions involving parentheses. When you have an expression like \(-3(x - 4)\), the distributive property tells us to multiply the term outside the parenthesis by each term inside the parenthesis. Specifically, \(-3\) is multiplied by both \(x\) and \(-4\). This process is known as "distribution."

• The result of distributing \(-3\) over \(x - 4\) is \(-3 \cdot x + (-3) \cdot (-4)\).
• Simplifying these multiplications gives us \(-3x + 12\).

By applying the distributive property, we've successfully transformed the complex expression into a simpler form that's easier to work with in solving the equation.

Equation Simplification

Simplifying equations is a critical skill in algebra that involves rearranging and combining like terms to make an equation easier to solve. In the exercise, after applying the distributive property, we obtained the equation \(-3x + 12 = -4x\). To simplify this equation:

• We rearrange the like terms by moving all terms with \(x\) to one side and constants to the other side.
• Add \(3x\) to both sides to isolate the \(x\) terms: \(12 = -4x + 3x\).

After simplification, you will have \(12 = -x\). This equation is now simplified to the point where one can directly solve for \(x\) by basic algebraic operations. Simplification helps remove clutter from the equation, making it more straightforward to find the solution.

Algebraic Verification

Algebraic verification is the process of checking your solution to ensure that it is correct. After solving for \(x\) and finding \(x = -12\), it is vital to verify this solution by substituting it back into the original equation. This step is crucial for confirming the accuracy of your solution.

• Plug \(x = -12\) into the original equation \(-3(x - 4) = -4x\).
• Calculate each side separately. For the left side: \(-3(-12 - 4) = -3(-16) = 48\).
• For the right side: \(-4(-12) = 48\).

Since both sides of the equation equal \(48\), the solution is verified as correct. Algebraic verification is like a safety net, ensuring that all your steps were done accurately and the solution is indeed correct.