Problem 12 - Algebra 1: Concepts and Skills

Question

Solve the equation. Check your solution in the original equation. $$ \frac{3}{8} t=6 $$

Step-by-Step Solution

Answer

The solution to the equation \(3(x+2)=4(5+x)\) is \(x = -14\)

Step 1: Expand the Parentheses

By applying the distributive property of multiplication over addition, the equation becomes \(3x + 6 = 20 + 4x\).

Step 2: Combine Like Terms

Subtract \(3x\) from both sides resulting in \(6 = 20 + x\). Then, subtract \(20\) from both sides of the equation, which simplifies the equation to \(x = -14\).

Step 3: Check Your Solution

Substitute \(x = -14\) back into the original equation \(3(x+2)=4(5+x)\). This yields \(3(-14 + 2) = 4(5 - 14)\), simplifying to \(-36 = -36\), confirming that the solution is correct.

Key Concepts

Distributive Property

Understanding the distributive property is essential in algebra, especially when it comes to solving linear equations. It states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. For example, when you see an expression like 3(x + 2), you apply the distributive property to 'distribute' the 3 to both x and 2.

In the equation 3(x + 2) = 4(5 + x), you would multiply 3 by x and 3 by 2, resulting in 3x + 6. Similarly, you would distribute the 4 on the right-hand side, resulting in 20 + 4x. This property helps to streamline equations by eliminating parentheses and setting the stage for further simplification through combining like terms.

Combining Like Terms

Once you've expanded equations using the distributive property, the next step is to combine like terms. Like terms are terms that contain the same variables raised to the same power. In the equation we're considering, after applying the distributive property, we have 3x + 6 = 20 + 4x.

To combine like terms, look for terms that have the same variable part. Here, 3x and 4x are like terms. By rearranging and subtracting 3x from both sides, you can group them together to eventually isolate the variable, leading to 6 = 20 + x, and after subtracting 20 from both sides, you get x = -14. It is important to perform the same operation on both sides of the equation to maintain its balance.

Checking Solutions in Algebra

The final step in solving an equation is to verify that your solution is correct. You do this by plugging the value you've found for the variable back into the original equation. In our example, the solution found was x = -14.

To check this, substitute -14 into x within the original equation, which gives you 3(-14 + 2) = 4(5 - 14). Simplifying both sides results in -36 = -36, demonstrating that both sides of the equation are equal when x = -14. This confirms that the solution is correct. Always remember to check your solutions, as this process will help you avoid errors and ensure your understanding of the equation's behavior.