Problem 28 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The solution is \(x = -6\).

Step 1: Write Down the Equation

First, write the given equation clearly: \[ \frac{1}{2}x - 3 = 6 + 2x \] This is the equation that we need to solve.

Step 2: Move All x Terms to One Side

Subtract \(2x\) from both sides to move all the \(x\) terms to one side:\[ \frac{1}{2}x - 2x - 3 = 6 \]

Step 3: Combine Like Terms

Combine the \(x\) terms on the left side:\[ \left(\frac{1}{2} - 2\right)x - 3 = 6 \]Simplifying, we get:\[ -\frac{3}{2}x - 3 = 6 \]

Step 4: Isolate the x Term

Add 3 to both sides to isolate the \(x\) term:\[ -\frac{3}{2}x = 9 \]

Step 5: Solve for x

Divide both sides by \(-\frac{3}{2}\) to solve for \(x\):\[ x = \frac{9}{-\frac{3}{2}} \] Simplify the division:\[ x = 9 \times \left(-\frac{2}{3}\right) = -6 \]

Step 6: Verify Algebraically

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

Step 7: Solve Graphically

Rewrite the equation by moving all terms to one side:\[ \frac{1}{2}x - 2x - 3 - 6 = 0 \]Simplify it to:\[ -\frac{3}{2}x - 9 = 0 \]Find the x-intercept of the line \( y = -\frac{3}{2}x - 9 \).Graph the line: it will intersect the x-axis at \(x = -6\).

Step 8: Conclusion

The solution to the equation is \(x = -6\), confirmed both algebraically and graphically.

Key Concepts

Algebraic Solution

Solving an equation algebraically involves manipulating the equation using algebraic operations such as addition, subtraction, multiplication, and division. The goal is to isolate the variable, often referred to as "solving for x." In the equation \( \frac{1}{2}x - 3 = 6 + 2x \), our task is to find the value of \( x \) that makes the equation true.

Here are the steps we took:

• We first shifted all the x terms to one side by subtracting \(2x\) from both sides. This helps consolidate all like terms together.
• Next, we combined the x terms: \( \frac{1}{2}x - 2x \) which simplifies to \( -\frac{3}{2}x \).
• By adding 3 to both sides, we isolated the variable term \( -\frac{3}{2}x = 9 \).
• The final step was to divide both sides by \( -\frac{3}{2} \) to get \( x = -6 \).

Algebraic solutions are very powerful as they provide a concrete answer and can be adapted to solve many types of equations efficiently.

Graphical Solution

The graphical solution of an equation involves plotting it on a graph to visually determine where it satisfies the condition of the equation being zero, usually where it crosses the x-axis. This method provides a different perspective from algebraic manipulation.

In solving the equation \( \frac{1}{2}x - 3 = 6 + 2x \) graphically, we interpret it as a linear equation. First, we rearrange it to the form \( y = -\frac{3}{2}x - 9 \).

Then, we plot the line:

• The slope of \( -\frac{3}{2} \) indicates the line descends steeply from left to right.
• The y-intercept is \(-9\), meaning the line crosses the y-axis at \( (0, -9) \).

By extending the line, we can find its x-intercept, the point where \( y = 0 \). This x-intercept represents the solution of the equation. Here, the line intersects the x-axis at \( x = -6 \). Therefore, \( x = -6 \) is the graphical solution of the equation, confirming our algebraic resolution.

Solving Equations

Solving equations is a fundamental aspect of algebra that involves finding the unknown variable that makes the equation true. Equations can be confronted using several methods, each bringing its unique advantages.

In the example provided, we used both algebraic and graphical approaches.

• Algebraic Method: This approach relies on systematic manipulation, using operations to simplify expressions and isolate variables.
• Graphical Method: Offers a visual understanding by examining where the graph of the equation intersects with the axes.

Each method enriches understanding: algebraic for precise value calculations and graphical for visual comprehension of relationships.
Deciding which method to use often depends on the context of the problem and the resources available, but mastering both can provide comprehensive skills in tackling various mathematical challenges.