Problem 78 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. $$2(5+6 x)=4(3 x-1)$$

Step-by-Step Solution

Answer

The equation has no solution as the simplified form led to a non-valid mathematical statement 0 = -14.

Step 1: Distribute

Distribute on both sides to expand the bracket. Hence, the equation becomes: \(2*5 + 2*6x = 4*3x - 4*1\). After performing the multiplication, it results in: \( 10 + 12x = 12x - 4\)

Step 2: Simplify

Simplify the equation by moving the variable to one side and constants to the other side. So, \(12x - 12x = - 4 - 10\). That results into \(0 = -14\).

Step 3: Check the Solution

There is no need to isolate for x as both sides of the equation result into a constant, hence having no solution.

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic concept that allows you to simplify expressions by eliminating parentheses. It states that multiplying a sum by a number is the same as multiplying each addend separately and then adding the results. In mathematical terms, this property is expressed as: \[ a(b + c) = ab + ac \]In the given equation, \(2(5 + 6x)\), we applied the distributive property. Here, the number 2 is distributed across both 5 and \(6x\), leading to the expanded expression \(2 \times 5 + 2 \times 6x\). Similarly, on the other side, \(4(3x - 1)\) becomes \(4 \times 3x - 4 \times 1\). Using this method ensures that all terms within the parentheses are accounted for, making the equation easier to solve in subsequent steps. This approach is especially useful when solving more complex algebraic equations where multiple terms are involved.

Simplifying Equations

Once you've distributed, the next step is simplifying the equation. Simplification involves combining like terms and isolating the variable. This process generally includes:

• Combining terms: Gather all similar terms on each side of the equation.
• Moving terms: Rearrange the equation to get all the variable terms on one side and constants on the other.

In our equation, \(10 + 12x = 12x - 4\), you'll notice that \(12x\) appears on both sides. By subtracting \(12x\) from both sides, we align terms properly. Thus, we effectively isolate constants, leading to \(0 = -14\), clearly indicating no solution as the equation is untrue.Simplifying is crucial as it clarifies whether an equation has a valid solution. In this example, our simplification proves that the given equation is inconsistent, meaning it cannot have any real solution.

Checking Solutions

Checking your solutions in algebra is a way to verify the results obtained from solving an equation. The process ensures your solution satisfies the original equation. However, some equations, like in our example with \(0 = -14\), show inconsistencies or contradictions upon simplification, meaning they have no solution. An equation with no solution is identified when two sides of the equation do not equal each other after simplification. There is no need to substitute back into the original equation for these cases because they have already been shown as invalid.In many cases, especially when solutions are present, substitution helps confirm that your answer is correct. Always re-evaluate your final results to validate or prove no solution. This step is vital in gaining assurance and understanding in algebraic problem-solving, allowing you to accurately approach more complex problems.