Problem 68 - College Algebra with Modeling and Visualization

Question

Use tables to solve the equation numerically to the nearest tenth. $$ 1-6 x=7 $$

Step-by-Step Solution

Answer

The solution to the equation \( 1 - 6x = 7 \) is \( x = -1 \).

Step 1: Rewrite the Equation

The equation given is \( 1 - 6x = 7 \). To simplify it, we start by moving the constant on the left side to the other side. Subtract 1 from both sides to get: \(-6x = 7 - 1\), which simplifies to \(-6x = 6\).

Step 2: Solve for x Numerically

Now, we solve for \( x \) by dividing both sides by -6. Thus, we have: \( x = \frac{6}{-6}\). We calculate the value: \( x = -1\).

Step 3: Verify the Solution

Substitute \( x = -1 \) back into the original equation to check: \( 1 - 6(-1) = 1 + 6 = 7 \), which is true. Hence, \( x = -1 \) is correct.

Key Concepts

Solving Equations

Solving equations is the process of finding the unknown variable that makes the equation true. It involves manipulating the equation using arithmetic operations to isolate the variable. In our given example, the equation is \( 1 - 6x = 7 \).

Here are the steps to solve such equations:

• Identify the variable: In this equation, \( x \) is the variable we need to solve for.
• Isolate the variable: Start by rearranging the equation to move all terms with the variable to one side. This often involves adding or subtracting terms from both sides. In our problem, we subtracted 1 from both sides to get \( -6x = 6 \).
• Solve for the variable: Once the variable is isolated, you typically divide or multiply both sides by a number to solve for the variable. Here, we divided both sides by \(-6\) to find \( x = -1 \).

After solving, it's always a good idea to check the solution by substituting it back into the original equation to ensure it holds true.

Numerical Solutions

Numerical solutions involve using approximations to find the solutions of equations when they cannot be solved easily by algebraic methods. These methods can often yield decimal or fractional answers.

In the equation \( 1 - 6x = 7 \), we found \( x = -1 \). Since the solution is exact and simple in this case, a numerical method ensured accuracy without further approximation. However, if the calculations resulted in more complex numbers or decimals, numerical solutions help in rounding off to required precision such as the nearest tenth.

Numerical methods are particularly useful for complex equations where the roots are not easy numbers. They provide a way to approximate solutions to any desired level of accuracy.

Tables in Algebra

Tables are a powerful tool in algebra that can help visualize and solve equations numerically. By creating a table of values, you can easily track and compare various outcomes.

While our problem \( 1 - 6x = 7 \) was straightforward to solve algebraically, tables often come in handy for more complex problems or to verify numerical solutions. In this context:

• Create a table with potential values of \( x \): List different values and calculate corresponding results to identify which agrees with the equation.
• Perform calculations: For each value, perform calculations to check if the left-hand side equals the right-hand side of the equation.
• Verify solutions: A correct \( x \) will yield a matching solution when substituted back into the equation.

Tables are especially useful when solutions require estimation or to track behavior of polynomial and complex functions visually.