Problem 13 - 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. $$x-5=6$$

Step-by-Step Solution

Answer

x = 11

Step 1: Isolate x

To solve for x, you need to isolate x on one side of the equation. Start by adding 5 to both sides of the equation to cancel the -5 on the left side.

Step 2: Simplify the Equation

After adding 5 to both sides of the equation, the left side becomes just x, and the right side becomes 11 (since 6 + 5 = 11).

Step 3: Write the Solution

Now that we have isolated x, we can write down the solution. x equals 11.

Step 4: Check the Solution

To check the solution, substitute x with 11 in the original equation and see if the equation holds true. So you would have 11 - 5, which equals 6. Since this is the same as the right side of the original equation, the solution is correct.

Key Concepts

Isolating Variables

Understanding how to isolate a variable is a foundational skill in solving linear equations. The goal is to get the variable by itself on one side of the equation. This is typically done by performing operations that reverse or 'undo' whatever is being done to the variable. For example, if a number is subtracted from the variable, you would add that number to both sides. In our exercise, we see that the variable x is decreased by 5, so we add 5 to both sides to isolate x. This process sets the stage for finding the solution to the equation.

Simplifying Equations

Once you've taken the first step to isolate the variable, the next step is to simplify the equation. This means performing the arithmetic operations to both sides of the equation to further refine the expression. In the given exercise, after adding 5 to both sides, the simplification step is simply 6 + 5, which yields 11. Keeping a neat and orderly simplification process will help avoid errors and make it easier to see the solution.

Checking Solutions

A crucial step that should never be overlooked is checking the solution you've found. This involves substituting the value back into the original equation to see if it holds true. For the exercise \(x-5=6\), we found that \((x = 11\). To check, we replace \((x\)) with 11 in the original equation, giving us \(11 - 5\), which simplifies to 6. Since this matches the original right side of the equation, we can confidently say the solution \(x = 11\) is correct. Checking confirms the reliability of our work and reinforces our understanding of the equation's dynamics.

Fractional Answers

When solving equations, especially in higher mathematics or when exact values are required, you may encounter fractional answers. These are exact representations, as opposed to decimal approximations, and are essential in precise calculations. To express answers as fractions, you must be familiar with numerator and denominator operations. Keeping the solution in fractional form can also make it easier to check the answer, as you're less likely to deal with the rounding errors that come with decimals. Moreover, having a handle on fractions strengthens the conceptual grasp of division and ratios in algebraic contexts.