Problem 2 - College Algebra with Modeling and Visualization

Question

Solve the rational equation (a) symbolically, (b) graphically, and (c) numerically $$ \frac{3 x}{2 x-1}=3 $$

Step-by-Step Solution

Answer

The solution is \( x = 1 \).

Step 1: Clear the equation of fractions

To solve the equation \( \frac{3x}{2x-1} = 3 \), we first eliminate the fraction by multiplying both sides by \( 2x - 1 \) (note we assume \( 2x-1 eq 0 \) to avoid division by zero). This gives us: \( 3x = 3(2x - 1) \).

Step 2: Expand and simplify the equation

Expand the right-hand side of the equation: \( 3x = 6x - 3 \).

Step 3: Rearrange the terms to isolate x

Subtract \( 6x \) from both sides to get: \( 3x - 6x = -3 \), resulting in \( -3x = -3 \).

Step 4: Solve for x

Divide both sides by \(-3\) to solve for \(x\), giving us: \( x = 1 \).

Step 5: Verify the solution graphically

To verify graphically, plot the function \( y = \frac{3x}{2x-1} \) and the line \( y = 3 \). The intersection of these graphs occurs at \( x = 1 \), confirming the solution matches the algebraic result.

Step 6: Verify the solution numerically

Substitute \( x = 1 \) back into the left-hand side of the original equation: \( \frac{3(1)}{2(1)-1} = 3 \). This reduces to \( \frac{3}{1} = 3 \), which is true, confirming \( x = 1 \) numerically.

Key Concepts

Solving Equations

When tackling rational equations, the key is to remove the fractions to simplify solving. In our given equation \( \frac{3x}{2x-1} = 3 \), we achieve this by multiplying both sides by the denominator \( 2x - 1 \). This step is crucial, as it clears the equation of fractions:

• Ensure \( 2x - 1 eq 0 \) to avoid division by zero.
• Multiply: \( 3x = 3(2x - 1) \), resulting in an equation without fractions.

Now, simplify and solve by expanding and rearranging terms. Start by expanding \( 3(2x - 1) \) to get \( 6x - 3 \). The new equation becomes \( 3x = 6x - 3 \). At this point, isolate \( x \) by bringing all \( x \) terms to one side:

• Subtract \( 6x \) from both sides: \( 3x - 6x = -3 \).
• Simplify to \( -3x = -3 \).

The final step is straightforward: solve for \( x \) by dividing both sides by \( -3 \) to find \( x = 1 \). Each manipulation brings us closer to finding \( x \) systematically, ensuring understanding and accuracy.

Graphical Methods

Graphical methods provide a visual means of verifying solutions to equations, which can be quite enlightening. To verify \( x = 1 \) graphically, we plot the rational function \( y = \frac{3x}{2x-1} \) and the line \( y = 3 \). The solution should correspond to the \( x \)-coordinate at which the two graphs intersect:

• Set up your graph with axes labelled.
• Plot \( y = \frac{3x}{2x-1} \), noting the shape and potential asymptotes.
• Draw the horizontal line \( y = 3 \), which is a constant.

Watch for where these two lines cross. The intersection point signifies where both expressions are equal, confirming the algebraic solution. For our specific case, at \( x = 1 \), both plotted graphs intersect exactly at this point on the graph. Visual confirmation like this can illuminate the relationships between different expressions and validate the solution from another perspective.

Numerical Verification

Numerical verification serves as an essential final check for the solution of a rational equation. By substituting the found solution back into the original equation, we can ensure it is indeed correct. Let’s see how this works for \( x = 1 \):

• Substitute \( x = 1 \) into the original equation: \( \frac{3(1)}{2(1) - 1} = 3 \).
• Simplify: \( \frac{3}{1} = 3 \).

The result is true, confirming the solution of \( x = 1 \) is accurate. This method is particularly useful when checking work or ensuring comprehension of the problem-solving process. Numerical verification provides a solid understanding that the solution logically fits the original equation. Beyond confirming correctness, this approach strengthens your grasp of how the equation behaves when \( x \) takes on various values, solidifying mathematical intuition.