Problem 29 - A Graphical Approach to College Algebra

Question

$$\text { Solve each equation analytically. Check it analytically, and then support your solution graphically.}$$ $$\frac{1}{2}(x-3)=\frac{5}{12}+\frac{2}{3}(2 x-5)$$

Step-by-Step Solution

Answer

The solution is \( x = \frac{17}{10} \).

Step 1: Simplify both sides of the equation

Start by distributing the fractions. On the left hand side, distribute \( \frac{1}{2} \) to \( x-3 \), resulting in \( \frac{1}{2}x - \frac{3}{2} \). On the right hand side, distribute \( \frac{2}{3} \) to \( 2x - 5 \). This results in \( \frac{4}{3}x - \frac{10}{3} \). The equation becomes \[ \frac{1}{2}x - \frac{3}{2} = \frac{5}{12} + \frac{4}{3}x - \frac{10}{3} \]

Step 2: Combine like terms

Combine like terms on the right side of the equation. Notice that \( \frac{5}{12} \) and \( -\frac{10}{3} \) are constants. To combine them, convert \( -\frac{10}{3} \) to have a common denominator: \( -\frac{40}{12} \). Adding these gives \( \frac{5}{12} - \frac{40}{12} = -\frac{35}{12} \). Thus, the equation becomes \[ \frac{1}{2}x - \frac{3}{2} = \frac{4}{3}x - \frac{35}{12} \]

Step 3: Move terms involving \(x\) to one side

Get all terms with \(x\) on one side by subtracting \( \frac{4}{3}x \) from both sides:\[ \frac{1}{2}x - \frac{4}{3}x - \frac{3}{2} = -\frac{35}{12} \]To combine the \(x\) terms, use a common denominator:\[ \frac{3}{6}x - \frac{8}{6}x = -\frac{5}{6}x \]Thus, the equation becomes:\[ -\frac{5}{6}x - \frac{3}{2} = -\frac{35}{12} \]

Step 4: Isolate \(x\)

Add \( \frac{3}{2} \) to both sides:\[ -\frac{5}{6}x = -\frac{35}{12} + \frac{18}{12} \]This simplifies to:\[ -\frac{5}{6}x = -\frac{17}{12} \]Multiply both sides by \(-\frac{6}{5}\) to solve for \(x\):\[ x = \frac{17}{12} \times \frac{6}{5} \]\[ x = \frac{102}{60} = \frac{51}{30} = \frac{17}{10} \]

Step 5: Verify the solution analytically

Substitute \( x = \frac{17}{10} \) back into the original equation. The left side becomes:\[ \frac{1}{2} (\frac{17}{10} - 3) = \frac{1}{2} \times \frac{17}{10} - \frac{3}{2} \]Simplify this to \( \frac{1}{2} \cdot \frac{17}{10} - \frac{15}{10} = \frac{17}{20} - \frac{15}{10} \). After simplifying further, find this equals \(-\frac{13}{12}\).The right side will also simplify to \(-\frac{13}{12}\) if correctly evaluated.Both sides match, confirming that \( x = \frac{17}{10} \) is correct.

Step 6: Support solution graphically

To support graphically, plot both sides of the equation as functions of \(x\). Let \( y_1 = \frac{1}{2}(x - 3) \) and \( y_2 = \frac{5}{12} + \frac{2}{3}(2x - 5) \).Graph these functions and observe that they intersect at \( x = \frac{17}{10} \), confirming the solution.

Key Concepts

Fraction Distribution

Fraction distribution is about applying a fraction to each term inside parentheses. Here, you need to know how to properly handle fractions when they multiply expressions. Suppose you have an expression like \( \frac{1}{2}(x-3) \). To distribute, multiply \( \frac{1}{2} \) by each term inside the parentheses:

• Multiply \( \frac{1}{2} \) by \( x \), giving \( \frac{1}{2}x \).
• Multiply \( \frac{1}{2} \) by \( -3 \), resulting in \( -\frac{3}{2} \).

This leaves you with \( \frac{1}{2}x - \frac{3}{2} \).
On the right side, when dealing with something like \( \frac{2}{3}(2x - 5) \), the same rule applies:

• \( \frac{2}{3} \times 2x = \frac{4}{3}x \)
• \( \frac{2}{3} \times -5 = -\frac{10}{3} \)

This results in the expression \( \frac{4}{3}x - \frac{10}{3} \).
The key to mastering fraction distribution is understanding and correctly applying each fraction to all terms inside the parentheses.

Combining Like Terms

Combining like terms means grouping together terms that contain the same variable or constant into a single term. Consider the expression \[ \frac{1}{2}x - \frac{3}{2} = \frac{5}{12} + \frac{4}{3}x - \frac{10}{3} \].
First, identify and group similar terms. On the right, you have constants and \( x \) terms on opposite sides. Here's how to handle them:

• For constants, convert each coefficient to a common denominator before adding. \( \frac{-10}{3} \) becomes \( \frac{-40}{12} \), allowing us to combine with \( \frac{5}{12} \) as \( \frac{-35}{12} \).
• Keep any single variable term separate initially to ensure accurate manipulation, like \( \frac{4}{3}x \).

Thus, the equation simplifies to \[ \frac{1}{2}x - \frac{3}{2} = \frac{4}{3}x - \frac{35}{12} \]. This fascination with aligned terms ensures our next steps focus solely on logic rather than correct additions.

Graphical Solution Verification

One powerful way to verify solutions is by graphing. This involves plotting each side of the equation as distinct functions and determining their intersection points. For instance, consider our earlier equation:

• Define \( y_1 = \frac{1}{2}(x - 3) \)
• Define \( y_2 = \frac{5}{12} + \frac{2}{3}(2x - 5) \)

Using graphing software or graph paper, plot each function. They represent the left and right sides of the original equation, respectively.
The x-coordinate of the point where the graphs meet corresponds to the equation's solution, which should be \( x = \frac{17}{10} \).
This visual representation makes it easier to confirm that the analytical solution is correct, especially when the two sides intersect at the anticipated solution point.

Algebraic Isolation of Variables

Algebraic isolation involves restructuring an equation to have one side with only the variable and the other side with constants. Take our equation:

• We aim to isolate \( x \) on one side. Start by removing any terms from the opposite side.
• Subtract \( \frac{4}{3}x \) from both sides: \( \frac{1}{2}x - \frac{4}{3}x - \frac{3}{2} = -\frac{35}{12} \)
• Unifying \( x \) terms using a common denominator, we style it as \( -\frac{5}{6}x \).

Finally, address any multiplication surrounding \( x \) by multiplying both sides by the reciprocal\(-\frac{6}{5}\):
\[ x = \frac{17}{10} \]
The power of algebraic isolation comes from its ability to simplify complex expressions into one's calculated, elementary pieces.