Problem 19 - Introductory Algebra for College Students

Question

Solve each equation. Be sure to check your proposed solution by substituting it for the variable in the original equation. $$3(5-x)=4(2 x+1)$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 1\).

Step 1: Distribute

Start by distributing on both sides of the equation, this gives: \(15 - 3x = 8x + 4\)

Step 2: Get all x terms on one side of the equation

Move the -3x from the left side to the right side by adding 3x to both sides, in order to remove the variable from the left side. This becomes: \(15 = 11x + 4\)

Step 3: Isolate x

Subtract 4 on both sides to get x by itself, yielding: \(11x = 11\)

Step 4: Solve for x

Divide both sides by 11 for the final solution: \(x = 1\)

Step 5: Verify solution

Substitute 1 back into the original equation to verify the solution: \(3(5 - 1) = 4(2*1 + 1)\). This simplifies to \(12 = 12\), verifying that our answer is correct.

Key Concepts

Solving Equations

To solve equations effectively, you need to isolate the unknown variable. This involves several systematic steps:

• First, distribute any necessary terms to get rid of parentheses. This means you expand any terms that are multiplied across a sum or difference.
• Second, make sure all terms involving the unknown variable are on one side of the equation, and constants are on the other side. You can do this by adding or subtracting terms on both sides of the equation.
• Finally, simplify the equation to solve for the unknown variable. This might include division or multiplication to isolate the variable further.

Each of these steps helps to transform the original equation into a simpler form, where the solution becomes apparent. By practicing these techniques, you'll improve your ability to solve equations efficiently.

Distribution Property

The distribution property is a useful algebraic principle. It states that when you multiply a number by a group of terms inside parentheses, you need to multiply the number by each term separately. The formula for the distribution property is: \[a(b + c) = ab + ac\]. This property helps to simplify equations before solving them. For example, in the given equation \(3(5-x)\), distribute \(3\) to both \(5\) and \(-x\). This results in \(15 - 3x\), making the equation simpler to work with. Using the distribution property is crucial in the solving process because it transforms complex equations into manageable ones, allowing you to focus on solving for variables and ultimately arriving at the solution.

Verifying Solutions

Once you find a potential solution to an equation, it's important to verify your answer. This ensures that the solution is correct and satisfies the original equation. Here’s how you can verify solutions:

• Substitute the found value back into the original equation to confirm both sides are equal.
• If the equation balances, the solution is correct.
• If it doesn't, you need to revisit your steps and identify any possible mistakes.

In the example provided, substituting \(x = 1\) into \(3(5 - x)\) and \(4(2x + 1)\) simplifies to \(12 = 12\). This confirms that \(x = 1\) is indeed the correct solution. Verifying solutions is a vital step to ensure accuracy in mathematical problem-solving.