Problem 22 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation analytically. Check it analytically, and then support the solution graphically. $$9 x-17=2 x+4$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 3\).

Step 1: Isolate the Variable

To begin solving the equation \(9x - 17 = 2x + 4\), start by isolating the variable on one side. Subtract \(2x\) from both sides to move all the terms containing \(x\) to one side of the equation: \[ 9x - 2x - 17 = 4 \] This simplifies to: \[ 7x - 17 = 4 \]

Step 2: Simplify the Equation

Now, isolate the constant term by adding \(17\) to both sides to remove the \(-17\) on the left side: \[ 7x = 4 + 17 \] This simplifies to: \[ 7x = 21 \]

Step 3: Solve for \(x\)

Next, solve for \(x\) by dividing both sides by \(7\) to get the variable alone: \[ x = \frac{21}{7} \] This reduces to: \[ x = 3 \]

Step 4: Check the Solution Analytically

Substitute \(x = 3\) back into the original equation to verify the solution: \[ 9(3) - 17 = 2(3) + 4 \] Calculate each side separately: - Left Side: \(27 - 17 = 10\) - Right Side: \(6 + 4 = 10\) Since both sides are equal, \(x = 3\) is correct.

Step 5: Graphical Support

To support the solution graphically, plot the two expressions \(y_1 = 9x - 17\) and \(y_2 = 2x + 4\) on a graph. They are linear equations, so they will be straight lines. The \(x\)-coordinate of the point where the lines intersect is the solution: - Plot \(y_1 = 9x - 17\). - Plot \(y_2 = 2x + 4\). The lines intersect at \((3, 10)\), confirming that \(x = 3\) is the solution.

Key Concepts

Isolate the Variable

In solving linear equations, one critical step is to isolate the variable. This means getting the variable by itself on one side of the equation. Let's consider the equation provided:
\( 9x - 17 = 2x + 4 \).Start by moving all terms containing the variable to the same side. We do this by subtracting \(2x\) from both sides:

• Subtract \(2x\) from both sides: \( 9x - 2x - 17 = 4 \).
• This simplifies to \( 7x - 17 = 4 \).

The next step is to move constant terms to the other side of the equation, so add \(17\) to both sides:- Adding \(17\) gives \( 7x = 21 \).Finally, solve for the variable by dividing by the coefficient of \(x\). Divide both sides by 7:
- This simplification gives \( x = \frac{21}{7} \), which reduces to \( x = 3 \).This method ensures the variable is isolated clearly to determine its value.

Graphical Solution

A graphical solution provides visual confirmation of the result. To verify our solution graphically, we can use the original linear equations and plot them on a coordinate plane.The given equations are:

• \( y_1 = 9x - 17 \)
• \( y_2 = 2x + 4 \)

Both equations represent straight lines in a 2D plane. The goal is to find where these two lines intersect.
- Plot the line for \( y_1 = 9x - 17 \): This line decreases steeply.- Plot the line for \( y_2 = 2x + 4 \): This line rises gently.On the graph, where these two lines cross is the solution to the equation, which occurs at the point \((3,10)\).
The \(x\)-coordinate of this intersection point confirms that \(x = 3\) as derived algebraically.

Analytical Check

An analytical check is a crucial step to ensure that the solution derived is indeed correct. This involves substituting the solution back into the original equation to verify consistency.Once solved, you substitute \(x = 3 \) back into the original expression:

• Left side: Substitute into \(9x - 17\): \[ 9(3) - 17 = 27 - 17 = 10 \].
• Right side: Substitute into \(2x + 4\): \[ 2(3) + 4 = 6 + 4 = 10 \].

Since both sides of the equation equal 10 when \(x = 3\), the analytical check confirms that our solution is accurate.
Doing an analytical check is important as it serves as a double-check against computational mistakes. This mechanism reinforces mathematical accuracy in finding solutions.