Problem 82 - College Algebra with Modeling and Visualization

Question

Solve the equation (to the nearest tenth) (a) symbolically, (b) graphically, and (c) numerically. $$ 5-8 x=3(x-7)+37 $$

Step-by-Step Solution

Answer

The solution is \( x = -1 \).

Step 1: Simplify both sides

First, expand and simplify the equation on the right side. The equation is given as: \[ 5 - 8x = 3(x - 7) + 37 \]Expand the parentheses on the right:\[ 5 - 8x = 3x - 21 + 37 \]Combine constant terms on the right:\[ 5 - 8x = 3x + 16 \]

Step 2: Collect like terms

Now, move all terms involving \(x\) to one side and constant terms to the other:\[ 5 - 16 = 3x + 8x \]Simplify both sides:\[ -11 = 11x \]

Step 3: Solve for x

Divide both sides by 11 to solve for \(x\):\[ x = \frac{-11}{11} \]\[ x = -1 \]

Step 4: Graphical Solution

To solve graphically, plot the functions \( y = 5 - 8x \) and \( y = 3(x - 7) + 37 \) on the same graph. Find the x-coordinate where the two functions intersect. The intersection point should be at \( x = -1 \).

Step 5: Numerical Solution

Use an iterative numerical method or calculator to approximate the solution to the equation \( 5 - 8x = 3(x - 7) + 37 \). Start with an initial guess and refine it. Through calculation, you will find \( x = -1 \) as the solution to the nearest tenth.

Key Concepts

Solving Equations

When solving equations, we often aim to find the value of the unknown variable that satisfies the equation. In the given equation, we have terms involving the variable \( x \), and our goal is to isolate \( x \) on one side of the equation. This means moving terms around and simplifying until we have \( x = \text{something} \).

In our example, we start by expanding expressions and combining like terms. Then, we move all terms containing \( x \) to one side and all constant terms to the other. After that, we can simplify and solve for \( x \).

• First, distribute any coefficients in front of parentheses.
• Next, combine like terms, which involve adding or subtracting the constants as needed.
• Finally, isolate the variable by performing algebraic operations such as addition, subtraction, multiplication, or division.

The solution found is \( x = -1 \). This means when \( x \) is \(-1\), both sides of the original equation equal the same value.

Graphical Methods

Graphical methods offer a visual way to solve equations, which can be really helpful for understanding how different functions interact. In this case, we have two expressions from each side of our original equation:

• \( y = 5 - 8x \)
• \( y = 3(x - 7) + 37 \)

To solve graphically, you would plot both equations on a graph with an \( x \)- and \( y \)-axis. Each line or curve represents an equation.

The solution to the equation is the \( x \)-value where the two graphs intersect. This is because at this point, their \( y \)-values are equal, satisfying the equation.

In our example, the graphs intersect at \( x = -1 \), confirming the solution found through algebraic methods.

Numerical Methods

Numerical methods provide another way to find approximate solutions to equations, often using technology or a calculator. This approach is particularly useful for more complex equations where symbolic or graphical solutions are hard to find.

One method is to use an iterative process, which involves making an initial guess and then refining that guess through calculations to get closer to the actual solution.

• Start with an initial guess for \( x \).
• Calculate the result of the equation with this guess.
• Adjust the guess based on whether the result is too high or too low, aiming for zero.

In our exercise, using this method also reveals \( x = -1 \) as the solution to the nearest tenth.

Symbolic Solutions

Symbolic solutions involve solving equations by manipulating symbols rather than numerical approximations. This is a classical algebraic approach. We use a series of known rules and operations to simplify the equation and find an exact answer.

Our equation is solved symbolically by:

• Expanding brackets to simplify terms.
• Combining like terms to make the equation easier to manage.
• Rearranging terms to isolate the variable.
• Dividing or multiplying as necessary to solve for the variable.

The advantage of symbolic solutions is their precision and clarity, leading us directly to \( x = -1 \) in our task. It's a reliable method when an exact solution is both possible and necessary.