Problem 32 - Precalculus Mathematics for Calculus

Question

Solve the equation both algebraically and graphically. $$\frac{1}{2} x-3=6+2 x$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -6\).

Step 1: Simplify the Equation

We start by moving all terms containing the variable \(x\) to one side of the equation. To do this, subtract \(\frac{1}{2}x\) from both sides: \[-3 = 6 + 2x - \frac{1}{2}x.\]This simplifies to: \[-3 = 6 + \frac{4}{2}x - \frac{1}{2}x,\]which further simplifies to: \[-3 = 6 + \frac{3}{2}x.\]

Step 2: Isolate the Variable Term

Next, subtract 6 from both sides to isolate the term \(\frac{3}{2}x\):\[-3 - 6 = \frac{3}{2}x.\]This results in:\[-9 = \frac{3}{2}x.\]

Step 3: Solve for the Variable

To solve for \(x\), multiply both sides by the reciprocal of \(\frac{3}{2}\), which is \(\frac{2}{3}\):\[x = -9 \cdot \frac{2}{3}.\]This simplifies to:\[x = -6.\]

Step 4: Verify Algebraic Solution

Substitute \(x = -6\) back into the original equation to verify:\[\frac{1}{2}(-6) - 3 = 6 + 2(-6).\]Simplify the left side: \[-3 - 3 = -6.\]Simplify the right side:\[6 - 12 = -6.\]Both sides equal \(-6\), so \(x = -6\) is correct.

Step 5: Solve Graphically - Plot the Functions

To solve graphically, rewrite the equation as two separate functions: \[y_1 = \frac{1}{2}x - 3\] and \[y_2 = 6 + 2x.\] Plot these functions on the same set of axes.

Step 6: Identify the Intersection Point

Determine where the graphs of \(y_1\) and \(y_2\) intersect. The intersection point represents the solution to the equation. The graphs intersect at the point \((-6, -6)\).Thus, the solution to \(\frac{1}{2}x - 3 = 6 + 2x\) is \(x = -6\).

Key Concepts

Algebraic Methods

Algebraic methods are tools for solving equations by manipulation of algebraic expressions. They help you isolate the variable you’re solving for, often through operations that maintain the equality of the equation, such as adding, subtracting, multiplying, or dividing both sides by the same number.
In our original exercise, we start by simplifying the equation: \(\frac{1}{2}x - 3 = 6 + 2x\). First, we move all the \(x\)-terms to one side, and constant terms to the other. This involves subtracting \(\frac{1}{2}x\) from both sides to combine like terms:

• Simplified equation: \(-3 = 6 + \frac{3}{2}x\)

After that, we isolate \(\frac{3}{2}x\) by subtracting 6 from both sides, which turns into:

• \(-9 = \frac{3}{2}x\)

Now, to solve for \(x\), multiply both sides by the reciprocal of \(\frac{3}{2}\) which is \(\frac{2}{3}\):

• Solution: \(x = -6\)

These steps systematically break down the equation until we find the solution, making algebraic methods a reliable way to solve linear equations.

Graphical Methods

Graphical methods involve plotting equations on a graph to find their intersections, which can represent solutions. It transforms an abstract equation into a visual form that might be easier to understand for some learners.
In our exercise, we looked at \(y_1 = \frac{1}{2}x - 3\) and \(y_2 = 6 + 2x\) to turn the original equation into graphical form. By plotting the graphs of these functions, we can visually determine where they intersect.

• The line \(y_1\) has a slope of \(\frac{1}{2}\) and an intercept of \(-3\).
• The line \(y_2\) has a slope of \(2\) and an intercept of \(6\).

By drawing both lines on the same graph, the point where they cross over is \((-6, -6)\), showing us that \(x = -6\) satisfies both equations. Graphical methods not only help in solving equations but also provide a visual insight into how functions relate to one another.

Verification of Solutions

Verification of solutions ensures that the derived answer truly satisfies the original equation. It's a necessary step to confirm that no mistakes were made during the solution process.
To verify, substitute the value found for \(x\) back into the original equation. In our case, we check \(x = -6\) for \(\frac{1}{2}(-6) - 3 = 6 + 2(-6)\). Simplifying both sides should give the same result:

• Left side: \(-3 - 3 = -6\)
• Right side: \(6 - 12 = -6\)
• Both sides equal \(-6\), verifying our solution is indeed correct.

Verification solidifies the answer, ensuring confidence in the outcome. It's a crucial step, especially in exams or real-world applications.

Intersection of Graphs

The intersection of graphs is a key concept in visualizing solutions to equations. When two graph lines intersect, they meet at a point that satisfies both equations simultaneously. This point of intersection essentially represents the solution to the system of equations at hand.
In our problem, we identified the functions \(y_1 = \frac{1}{2}x - 3\) and \(y_2 = 6 + 2x\). By plotting these two linear equations on the same set of axes, we found they intersected at \((-6, -6)\).

• This point means that for \(x = -6\), both functions reach \(y = -6\), thus satisfying both equations.

Understanding intersections can also apply to systems with more variables or non-linear equations. It bridges the gap between algebraic solutions and graphical interpretations, bringing another layer of understanding of how equations work.