Problem 29 - 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. $$12(x+2)=15(x-4)-3$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 29\)

Step 1: Distribute Numbers

Distribute the numbers on both sides of the equation: \(12x + 24 = 15x -60 -3\)

Step 2: Collect Like Terms

Now, subtract 15x from both sides and also subtract 24 from both sides to get the equation in the form \(f(x)=0\): \( -3x + 87 = 0 \)

Step 3: Solve for x

Add 3x to both sides and then divide both sides by 3 to get the solution: \(x = 29\)

Step 4: Verify Solution With a Graphing Utility

This step involves graphing the function \(f(x) = -3x + 87\) and checking that it intersects the x-axis at \(x = 29\). Since a graph cannot be included in this json response, it is only stated here what you would do. A proper graphing program must be used for completion of this step.

Key Concepts

Distributive Property

The distributive property is an important concept in algebra, allowing you to simplify expressions. It states that any term multiplied by terms inside parentheses must be distributed, or multiplied, individually. In the example equation \( 12(x+2) = 15(x-4) - 3 \), the distributive property is used to multiply 12 by \( x+2 \) and 15 by \( x-4 \). Here’s how it's applied:

• Left Side: \( 12(x+2) \) becomes \( 12x + 24 \).
• Right Side: \( 15(x-4) - 3 \) becomes \( 15x - 60 - 3 \).

This simplifies the equation, setting the stage for solving it.

Like Terms

After using the distributive property, it’s crucial to combine like terms. Like terms are terms that include the same variables raised to the same power, making it possible to consolidate them into a single term. In the example of \( 12x + 24 = 15x - 63 \), combining like terms is the next logical step.

• X terms: On the left, you have \( 12x \) and on the right, \( 15x \). Subtract 15x from both sides to isolate the variable, resulting in \( 12x - 15x = -3x \).
• Constant terms: On the left, there are no constants apart from 24, while on the right, \(-60 -3 = -63\). Subtract 24 from both sides to simplify further, resulting in \( -63 - 24 = -87 \).

The rearrangement gives you \( -3x + 87 = 0 \), an equation ready for further simplification.

Algebraic Solution

Finding an algebraic solution involves isolating the variable on one side of the equation. The derived equation \( -3x + 87 = 0 \) leads to finding the value of \( x \).

• First, add \( 3x \) to both sides to obtain \( 87 = 3x \).
• Next, divide both sides by 3, providing the solution \( x = 29 \).

This step-by-step isolation of the variable is central to solving the equation algebraically and results in an understandable, concise answer.

Graphing Utility

Once you have an algebraic solution, a graphing utility can verify the correctness of your solution visually. The benefit of graphing is providing a visual representation of where the equation equals zero:

• Start by rewriting the equation in the form \( f(x) = 0 \). Here, it becomes \( f(x) = -3x + 87 \).
• Using a graphing tool, like Desmos or a graphing calculator, plot \( f(x) \).
• Check that the graph intersects the x-axis at \( x = 29 \), confirming the solution \( x=29 \).

Graphing serves as an effective method to double-check algebraic solutions by providing a different, clear perspective.