Problem 36 - 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 . $$ -5(x+1)+4(2 x-3)=2(x+2)-8 $$

Step-by-Step Solution

Answer

The solution is \(x = 13\).

Step 1: Simplify Each Side of the Equation

First, we need to distribute the numbers outside of the parentheses on each side of the equation. Start with the left side: \(-5(x + 1)\) becomes \(-5x - 5\).Then, distribute in \(4(2x - 3)\):\(4 \times 2x = 8x\)\(4 \times -3 = -12\)Thus, the left side simplifies to \(-5x - 5 + 8x - 12\).Now for the right side: Distribute in \(2(x + 2)\):\(2 \times x = 2x\)\(2 \times 2 = 4\)The right side simplifies to \(2x + 4 - 8\).

Step 2: Combine Like Terms

Combine the like terms: **Left Side:**Combine \(-5x + 8x\) to get \(3x\), and combine \(-5 - 12\) to get \(-17\).Thus, the left side becomes \(3x - 17\).**Right Side:**Combine \(4 - 8\) to get \(-4\).Thus, the right side becomes \(2x - 4\).

Step 3: Set the Simplified Equation

Now, the equation is simplified to:\(3x - 17 = 2x - 4\).

Step 4: Solve for x

Isolate \(x\) by moving all terms involving \(x\) to one side. Subtract \(2x\) from both sides:\(3x - 2x - 17 = -4\)This simplifies to:\(x - 17 = -4\)Add \(17\) to both sides to solve for \(x\):\(x = 13\).

Step 5: Check the Solution

Substitute \(x = 13\) back into the original equation to check: Left side: \(-5(13 + 1) + 4(2 \times 13 - 3)\) becomes:\(-5 \times 14 + 4 \times 23\)= \(-70 + 92\)= 22Right side: \(2(13 + 2) - 8\) becomes:\(2 \times 15 - 8\)= \(30 - 8\)= 22Since both sides are equal when \(x = 13\), our solution is correct.

Key Concepts

Distributive Property

When solving linear equations that contain parentheses, like \(-5(x+1)+4(2x-3)\), the distributive property is a crucial first step. This property allows us to eliminate the parentheses by distributing, or multiplying, the factor outside the parentheses by each term inside the parentheses. This can be represented by the formula: \(a(b + c) = ab + ac\).

For example, looking at the term \(-5(x + 1)\), the number \(-5\) multiplies by both \(x\) and \(1\), giving us \(-5x - 5\). Likewise, for \(4(2x - 3)\), the \(4\) multiplies with \(2x\) and \(-3\), resulting in \(8x - 12\).

• Apply distributive property to unravel the equation.
• Clear all the parentheses before proceeding with the next steps.

Remember, the distributive property helps simplify your equation, which sets the stage for combining like terms and further simplifying.

Combining Like Terms

After using the distributive property, the next important step in simplifying a linear equation is combining like terms. This means you need to gather all terms involving the same variable together, and separately combine constant terms. Doing so makes the equation easier to solve.

On the left side of the equation, for instance, after simplification, we have \(-5x - 5 + 8x - 12\). We combine similar terms:

• Combine the terms involving \(x\): \(-5x + 8x\) becomes \(3x\).
• Combine the constant terms: \(-5 - 12\) becomes \(-17\).

This turns the left side of the equation into \(3x - 17\).

For the right side, \(2x + 4 - 8\), combining constants gives: \(4 - 8 = -4\), making it become \(2x - 4\).
This process of combining like terms simplifies the equation, making it much more manageable to solve.

Checking Solutions

After simplifying the equation and solving for the variable, it’s crucial to check your solution. Substituting the value back into the original equation ensures your solution is correct and consistent.

In our example, after solving \(3x - 17 = 2x - 4\), we found \(x = 13\). To verify this:

• For the left side, substitute \(x = 13\) into \(-5(13 + 1) + 4(2 \times 13 - 3)\).
• Calculate: \(-5 \times 14 + 4 \times 23 = -70 + 92 = 22\).

• For the right side, substitute again into \(2(13 + 2) - 8\).
• Calculate: \(2 \times 15 - 8 = 30 - 8 = 22\).

Both sides are equal when \(x = 13\), confirming that the solution is correct. This step is essential in ensuring that your solution fully satisfies the original equation and hasn't overlooked any potential errors.