Problem 3 - Technical Mathematics with Calculus

Question

Solve and check each equation. Treat the constants in these equations as exact numbers. Leave your answers in fractional, rather than decimal, form. $$30+5 x=20 x$$

Step-by-Step Solution

Answer

x = 2

Step 1: Move the variable terms to one side

Subtract 5x from both sides of the equation to get the variable terms on one side. This gives us 30 = 20x - 5x.

Step 2: Combine like terms

Combine the x terms on the right side of the equation to simplify. This results in 30 = 15x.

Step 3: Solve for x

Divide both sides of the equation by 15 to solve for x. This yields x = \(\frac{\text{30}}{\text{15}}\).

Step 4: Simplify the fraction

Simplify the fraction on the right side of the equation to find the value of x. So x = 2.

Step 5: Check the solution

Plug x = 2 back into the original equation to verify the solution: 30+5(2)=20(2). Simplify both sides to see if they are equal: 40 = 40, which is a true statement.

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a cornerstone technique in solving equations and finding the value of unknowns. It involves applying the rules of arithmetic to both sides of an equation to maintain equality while isolating the variable. In the process of solving the equation \(30 + 5x = 20x\), the first step is crucial: moving the variable terms to one side. By subtracting \(5x\) from both sides, we are manipulating the equation to facilitate further simplification. This step adheres to the law of equality, which states that what you do to one side of an equation, you must also do to the other to keep the equation balanced.

It is through algebraic manipulation that the stage is set for combining like terms and eventually solving for the variable. Mastery of algebraic manipulation – including operations like addition, subtraction, multiplication, and division applied judiciously – is essential for solving not just simple linear equations but also more complex algebraic expressions.

Combining Like Terms

Combining like terms is another fundamental aspect of solving linear equations. It simplifies the equation, making it easier to solve. Like terms are terms that contain the same variable raised to the same power. In the step by step solution, after the algebraic manipulation in step 1, step 2 involves combining like terms to further simplify the equation. The right side of the equation from step 1, \(20x - 5x\), demonstrates like terms because both contain the variable 'x'.

By combining them, we obtain \(15x\), thereby condensing the equation to \(30 = 15x\). This process of combining like terms not only simplifies calculations but also brings us closer to isolating the variable 'x', setting us up for the next step of the solution. Remember, when we combine like terms, we add or subtract the coefficients and keep the variable part unchanged.

Checking Solutions in Algebra

Checking your solutions is the final, but equally important, step in solving algebraic equations. After isolating the variable and simplifying to find that \(x = 2\), how do we know this is correct? We substitute the found value back into the original equation to ensure both sides are equal. For our example, plugging \(x = 2\) into the original equation \(30 + 5x = 20x\) yields \(40 = 40\), confirming that both sides of the equation are indeed equal.

This validation step not only confirms the correctness of our solution but also enhances our understanding of the relationship between the variables and constants in the equation. It serves as a proof that our algebraic manipulations and combination of like terms were executed correctly. Always remember to check your solutions to ensure accuracy and to gain confidence in your algebraic proficiency.