Problem 33 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation analytically. Check it analytically, and then support the solution graphically. $$0.40 x+0.60(100-x)=0.45(100)$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 75 \).

Step 1: Simplify the Equation

Start by distributing the constants to the variables and numbers within the parentheses. The equation is given as:\[ 0.40x + 0.60(100 - x) = 0.45(100) \]Distribute the 0.60 over the terms in the parenthesis:\[ 0.40x + 60 - 0.60x = 45 \]

Step 2: Combine Like Terms

After distributing, simplify the equation by combining like terms. In this case, combine the terms with \( x \):\[ (0.40x - 0.60x) + 60 = 45 \]This simplifies to:\[ -0.20x + 60 = 45 \]

Step 3: Isolate the Variable

Move the constant term to the other side of the equation to isolate the \( x \) term. Subtract 60 from both sides:\[ -0.20x = 45 - 60 \]This gives us:\[ -0.20x = -15 \]

Step 4: Solve for x

Divide both sides of the equation by -0.20 to solve for \( x \):\[ x = \frac{-15}{-0.20} \]Simplifying gives:\[ x = 75 \]

Step 5: Check the Solution Analytically

Substitute \( x = 75 \) back into the original equation to ensure correctness:\[ 0.40(75) + 0.60(100 - 75) = 0.45(100) \]Calculate each part:\[ 30 + 0.60(25) = 45 \]\[ 30 + 15 = 45 \]Since both sides are equal, the solution \( x = 75 \) is correct.

Step 6: Graphical Representation

Plot the left side of the equation \( y = 0.40x + 0.60(100-x) \) and the right side \( y = 0.45(100) \) which equals 45, each as separate lines on a graph. The intersection point of the two lines at \( x = 75 \) confirms that the solution is correct, as both lines intersect exactly once at this point, showing the equality.

Key Concepts

Analytical Solutions

Analytical solutions are a methodical means of solving equations by manipulating algebraic expressions to find values of unknown variables. In this exercise, the goal is to solve the equation \(0.40x + 0.60(100-x) = 0.45(100)\) by systematically using algebraic operations to isolate the required variable, \(x\).

To start, we distribute the constants across the parenthetical expression, making the equation easier to handle. This involves performing operations such as distributing and combining like terms.

Once simplified, the equation becomes \(-0.20x + 60 = 45\), which is more straightforward.

From this point, the variable \(x\) becomes more visible as we isolate it by subtracting terms and eventually dividing, giving us a clear solution: \(x = 75\).

This systematic approach helps in solving equations exactly, and secures a mathematically sound solution.

Graphical Method

The graphical method provides a visual approach to solving equations and verifying solutions. It complements the analytical method by translating the equation into a graphical representation on the coordinate plane. For the task at hand, we graph the two sides of the equation separately.

In doing this, the left side, \(y = 0.40x + 0.60(100-x)\), and the right side, \(y = 0.45(100)\), are plotted as lines.

The intersection of these two lines is key; this point represents the solution to the equation. In the example provided, the intersection occurs precisely at \(x = 75\), validating our analytical result.

By using the graphical method, we benefit from a clear visual cue and verify the consistency and correctness of our analytical findings.

Equation Simplification

Equation simplification is a crucial aspect of solving algebraic problems. It involves making an equation easier to work with through various algebraic techniques. Often, this starts with distributing terms within parentheses, as seen in the given equation, \(0.40x + 0.60(100-x) = 0.45(100)\).

This distribution helps break down complex expressions into more manageable parts. After distribution, combining like terms further simplifies the equation. For example, terms involving \(x\) are combined to form a single term such as \(-0.20x\).

Finally, solving for the variable may involve redeveloping the equation into a simpler form, such as \(-0.20x + 60 = 45\). This step by step simplification process is essential as it precisely isolates the variable, ensuring an accurate solution. Simplification lays the foundation for a clear pathway to solve equations efficiently.