Problem 27 - A Graphical Approach to College Algebra

Question

$$\text { Solve each equation analytically. Check it analytically, and then support your solution graphically.}$$ $$\frac{2 x+1}{3}+\frac{x-1}{4}=\frac{13}{2}$$

Step-by-Step Solution

Answer

The solution is \( x = 7 \).

Step 1: Eliminate Fractions

To eliminate the fractions, find the least common multiple (LCM) of the denominators 3, 4, and 2. The LCM is 12. Multiply each term in the equation by 12 to clear the fractions:\[ 12 \left( \frac{2x+1}{3} \right) + 12 \left( \frac{x-1}{4} \right) = 12 \times \frac{13}{2} \]This simplifies to:\[ 4(2x+1) + 3(x-1) = 78 \].

Step 2: Distribute and Simplify

Distribute the constants across the terms in the parentheses:\[ 8x + 4 + 3x - 3 = 78 \]Combine like terms:\[ 11x + 1 = 78 \].

Step 3: Isolate the Variable

Subtract 1 from both sides to isolate the term containing \( x \):\[ 11x = 77 \].

Step 4: Solve for the Variable

Divide both sides by 11 to solve for \( x \):\[ x = 7 \].

Step 5: Verify the Solution Analytically

Substitute \( x = 7 \) back into the original equation to verify:\[ \frac{2(7) + 1}{3} + \frac{7 - 1}{4} = \frac{13}{2} \]Simplifying the left side:\[ \frac{15}{3} + \frac{6}{4} = 5 + 1.5 = 6.5 \]Calculating \( \frac{13}{2} \), we get 6.5, hence the solution \( x = 7 \) matches.

Step 6: Graphical Representation

Graph the functions \( y_1 = \frac{2x+1}{3} + \frac{x-1}{4} \) and \( y_2 = \frac{13}{2} \) on the same plot. The x-value where they intersect is the solution.Plotting these, the lines intersect at the point \( x = 7 \), confirming our solution.

Key Concepts

Fractions in Equations

When working with algebraic equations, fractions can often seem tricky. However, they can be simplified easily with the right approach. Fractions in equations are usually tackled by first finding a common denominator. This helps to eliminate the fractions altogether, making the equation simpler to work with.

• The Least Common Multiple (LCM) is used to find a common denominator for all terms.
• Multiply each term by the LCM to eliminate fractions.

This technique allows you to transform a complex-looking equation into one that is more straightforward. For the exercise, the LCM of the denominators 3, 4, and 2 was 12. Multiplying each term by 12 cleared the fractions:\[ 12 \left( \frac{2x+1}{3} \right) + 12 \left( \frac{x-1}{4} \right) = 12 \times \frac{13}{2} \]Converting fractions into whole numbers simplifies calculations and reduces errors.

Graphical Representation of Solutions

Graphical representation is an intuitive way to understand and validate solutions to algebraic equations. By plotting the equations on a graph, you can visually interpret the point of intersection as the solution.In the context of the exercise, we use the functions:

• \( y_1 = \frac{2x+1}{3} + \frac{x-1}{4} \)
• \( y_2 = \frac{13}{2} \)

When graphed, the intersection point indicates the solution where both equations have the same value. Here, the intersection occurs at \( x = 7 \). Using graphs not only confirms analytical solutions but also provides insights into the behavior of the functions involved. It clearly displays how each function behaves over a range of x-values, helping in understanding the nature of the solution.

Solving Linear Equations

Solving linear equations involves isolating the variable, often achieved by straightforward algebraic manipulations. This step-by-step approach ensures clarity and accuracy.The process generally includes:

• Distributing any constants across terms.
• Combining like terms.
• Isolating the variable on one side of the equation.
• Solving for the variable by performing inverse operations, like division or subtraction.

For example, in the given exercise, after distributing and simplifying, the equation:\[ 8x + 4 + 3x - 3 = 78 \]Simplified to:\[ 11x + 1 = 78 \]From there, isolating \( x \) by subtracting 1 and dividing by 11 reveals \( x = 7 \). It's crucial not only to find the solution but also to verify it by substituting back into the original equation. This ensures the solution is consistent and accurate. By following these steps, solving linear equations becomes a reliable and straightforward process.