Problem 28 - 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 . $$ -2(x-1)=-3 x $$

Step-by-Step Solution

Answer

The solution is \( x = -2 \).

Step 1: Distribute the Coefficient

Start by distributing the -2 across the terms in the parentheses on the left side. This requires multiplying both terms inside the parentheses by -2.\[-2(x-1) = -2 \cdot x + 2 \cdot 1 = -2x + 2\]

Step 2: Simplify the Equation

After distributing, the equation is simplified to:\[-2x + 2 = -3x\]

Step 3: Combine Like Terms

To solve for \( x \), bring all terms involving \( x \) to one side of the equation and constants to the other. Add \( 2x \) to both sides:\[-2x + 2 + 2x = -3x + 2x\]This simplifies to:\[2 = -x\]

Step 4: Solve for x

Now, solve for \( x \) by multiplying both sides by \(-1\):\[-2 = x\]

Step 5: Check the Solution

Substitute \( x = -2 \) back into the original equation to ensure it's correct:\[-2(-2 - 1)\stackrel{?}{=}-3(-2)\]Simplify both sides:Left side: \[-2(-3) = 6\]Right side: \[6\]Both sides are equal, confirming that \( x = -2 \) is indeed the solution.

Key Concepts

The Distributive Property in Solving Equations

One of the key concepts when solving linear equations involving parentheses is the distributive property. This property allows us to eliminate the parentheses by distributing the multiplication over addition or subtraction within the parentheses. For instance, if you have an expression like

• \(-2(x-1)\),

you apply the distributive property by multiplying \(-2\) with each term inside the parentheses. This means multiplying \(-2\) by \(x\), and \(-2\) by \(-1\). Here’s how it looks:

• \(-2 \cdot x + (-2) \cdot (-1) = -2x + 2\)

Breaking it down step-by-step ensures less room for errors. Remember that the order of operations is crucial; distribute multiplication first, then proceed to other operations like combining terms.

Combining Like Terms to Simplify

After distributing, the next step is to make the equation simpler by combining like terms. Like terms are the terms that contain the same variable raised to the same power. In the simplified equation \(-2x + 2 = -3x\), you see terms involving \(x\) on both sides. The goal is to bring all \(x\)-related terms on one side and constant terms on the other.

• Add \(2x\) to both sides to move all terms with \(x\) together:
• \(-2x + 2 + 2x = -3x + 2x\)
• This simplifies to: \(2 = -x\)

By combining like terms, you are essentially clearing the clutter from the equation, which makes it easier to isolate the variable. This is a critical step because it paves the way to solving for the unknown in just a few more moves.

Checking Solutions for Accuracy

Once you solve for the variable, it's essential to verify your solution through a process called checking solutions. This ensures that your solution satisfies the original equation from start to finish. For the given problem, we solved for:

• \(x = -2\)

To check this, substitute \(-2\) back into the original equation:

• \(-2(-2-1)\stackrel{?}{=}-3(-2)\)

Simplify both sides:

• Left side: \(-2(-3) = 6\)
• Right side: \(-3(-2) = 6\)

Since both the left and right sides of the equation are equal, the solution \(x = -2\) is confirmed as correct. Checking your solution is like a final safety net—it helps catch any possible mistakes made during the calculations.