Problem 10 - 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. $$4 x=3 x-6$$

Step-by-Step Solution

Answer

The solution to the equation is x = -6. After checking, the solution satisfies the original equation.

Step 1: Isolate the variable

To solve the equation, first, we need to isolate the variable x on one side of the equation. We do this by subtracting 3x from both sides of the equation to eliminate it from the right side. This results in the equation 4x - 3x = -6.

Step 2: Simplify the equation

Next, we simplify the left side of the equation by combining like terms. The equation 4x - 3x simplifies to x, giving us the simplified equation x = -6.

Step 3: Check the solution

To check the solution, we substitute the value of x back into the original equation and see if both sides are equal. Substituting x with -6, we get 4(-6) = 3(-6) - 6, which simplifies to -24 = -18 - 6. Simplifying the right side, we get -24 = -24, which is true, confirming that x = -6 is the correct solution.

Key Concepts

Isolating the Variable

The process of solving linear equations often starts with isolating the variable. This means to get the variable on one side of the equation all by itself. To do this, use inverse operations to move other terms that contain the variable to the opposite side. For example, if you start with the equation 4x = 3x - 6, you want to isolate x. To accomplish this, subtract 3x from both sides, getting 4x - 3x = -6. After performing the subtraction, x stands alone on one side of the equation, indicating that it has been successfully isolated.

It is important to always perform the same operation on both sides of the equation to maintain the balance. If you imagine the equation as a scale, whatever you do to one side, do to the other to keep it level. This step is crucial because it sets the stage for solving the equation, by ensuring that the variable is unaccompanied and ready to be identified.

Simplifying Equations

Once the variable is isolated, simplifying the equation is the next essential step. This involves combining like terms and reducing expressions to their simplest form. In the context of our example, 4x - 3x can be simplified because both terms involve the variable x. Combining them, we simply calculate 4 - 3, which equals 1, so the equation simplifies to x = -6.

Simplification might also involve reducing fractions, distributing products, or combining constants. Each of these actions simplifies the expression and brings you closer to the solution. Simplification makes the equation more concise and manageable, ultimately leading to a clear solution. Remember, a simplified equation is the clearest path to an accurate solution and minimizes potential errors in calculation.

Checking Solutions

After finding a potential solution to an equation, it is essential to verify its correctness. This is done by checking the solution, which involves substituting the value back into the original equation. For the example at hand, where the proposed solution is x = -6, we substitute -6 for x in the original equation, 4x = 3x - 6, yielding 4(-6) = 3(-6) - 6. Now simplify the equation to see if both sides equal the same value.

After simplification, -24 = -24 confirms that our solution is correct, because the result is a true statement. This verification step is crucial—it ensures that no mistake was made during the isolation and simplification steps. Always remember to check your solutions to validate your results. It's a fundamental practice that not only affirms your answer but also reinforces your understanding of the concepts involved in solving linear equations.