Problem 34 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation algebraically. Then write the equation in the form \(f(x)=0\) and use a graphing utility to verify the algebraic solution. $$0.75 x+0.2(80-x)=3.9$$

Step-by-Step Solution

Answer

The solution to the equation \(0.75 x + 0.2(80 - x) = 3.9\) is \(x = -12.1 / 0.55\). The function form of the equation to be graphed is \(f(x) = 0.55x + 16 - 3.9\). The solution can be verified graphically by checking if \(f(x) = 0\) at \(x = -12.1 / 0.55\).

Step 1: Simplify the Equation

The equation to solve is \(0.75 x + 0.2(80 - x) = 3.9\). First, simplify the right-hand side of the equation, this yields \(0.75x + 16 - 0.2x = 3.9\).

Step 2: Combine like terms

We combine like terms on the left side of the equation to get \(0.55x + 16 = 3.9\).

Step 3: Isolate the x variable

We must now isolate the \(x\) variable. Subtract 16 from both sides which will give \(0.55x = -12.1\).

Step 4: Solve for x

Finally, to isolate \(x\), divide both sides by 0.55. This gives \(x = -12.1/0.55\). Solving this will yield the solution for \(x\).

Step 5: Convert the equation to f(x) = 0 form

The function form of the equation will be \(f(x) = 0.55x + 16 - 3.9\). This is the function which will be graphed with a graphing utility.

Step 6: Verify the solution graphically

To confirm that our solution is correct, plot the function \(f(x) = 0.55x + 16 - 3.9\), and check to see whether \(f(x) = 0\) at the solution found earlier.

Key Concepts

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. In the equation \(0.75x + 0.2(80-x) = 3.9\), the goal is to determine what value of \(x\) satisfies this relationship. First, simplify the equation by distributing and combining like terms:

• Distribute: \(0.2(80-x)\) to get \(16 - 0.2x\).
• Combine like terms: \(0.75x - 0.2x\) becomes \(0.55x\).

This yields a simpler equation: \(0.55x + 16 = 3.9\). By subtracting 16 from both sides and solving for \(x\), you find the value that balances the equation, ultimately solving it.

Graphing Utility

A graphing utility is a tool that helps visualize equations and their solutions. Once you have an equation like \(f(x) = 0\), a graphing utility can provide a graphical representation. You can observe where the graph intersects the x-axis, confirming your algebraic work is accurate. Graphing utilities include:

• Graphing calculators
• Online graphing software
• Mathematical software packages

By plotting the simplified function \(f(x) = 0.55x + 16 - 3.9\), you can visually confirm where \(f(x) = 0\). The x-coordinate of this intersection point should match the solution found algebraically.

Function Form

Writing an equation in function form involves rearranging it into the format \(f(x) = 0\). For the equation above, this means expressing it as \(f(x) = 0.55x + 16 - 3.9\). This form is immensely useful in understanding and analyzing the behavior of the function.

• Makes graphing straightforward.
• Clearly identifies the equation as a function.
• Facilitates using graphing utilities.

By setting the function equal to zero, you can easily find the roots, which correspond to the solutions of the equation. It’s an essential step when moving from algebraic solutions to graphical analysis.

Algebraic Solution

An algebraic solution finds the exact value of the variable using mathematical operations. For \(0.55x + 16 = 3.9\), algebraic manipulation leads to a precise value of \(x\). Steps to achieve this include:

• Isolating the variable \(x\) on one side of the equation.
• Performing inverse operations to unravel the equation.
• Dividing by the coefficient \(0.55\) to solve for \(x\).

This process ensures that you determine the precise numerical value for \(x\) that makes the original equation true, which can then be checked graphically for confirmation.