Problem 21 - A Graphical Approach to College Algebra

Question

$$\text { Solve each equation analytically. Check it analytically, and then support your solution graphically.}$$ $$2 x-5=x+7$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 12\).

Step 1: Move Variables to One Side

Start by moving all the variable terms to one side of the equation. Subtract \(x\) from both sides:\[2x - 5 - x = x + 7 - x\]Simplifies to:\[x - 5 = 7\]

Step 2: Solve for the Variable

Isolate the variable \(x\) by adding 5 to both sides of the equation:\[x - 5 + 5 = 7 + 5\]This simplifies to:\[x = 12\]

Step 3: Verify the Solution

Substitute \(x = 12\) back into the original equation to ensure it satisfies the equation:\[2(12) - 5 = 12 + 7\]which becomes:\[24 - 5 = 19\]Thus, \[19 = 19\], confirming that the solution is correct.

Step 4: Graphical Support

Graph the functions \(y = 2x - 5\) and \(y = x + 7\). The solution \(x = 12\) can be visually verified as the point where these two lines intersect. This intersection should occur at the point (12, 19), confirming our analytical solution.

Key Concepts

Graphical Verification

Using graphs is a great way to visually verify the solution of a linear equation. To do this, we plot the two expressions that make up the sides of the equation as separate lines on a coordinate plane.
In our example, we have the equations:

• \( y = 2x - 5 \)
• \( y = x + 7 \)

By graphing these, you can clearly see where the two lines intersect. This intersection point represents the solution to the equation. For our problem, the lines intersect at the point (12, 19).
This graphical method offers a visual representation, allowing us to confirm that when \( x = 12 \), both equations yield a resulting \( y \) value of 19. Thus, graphs serve as a helpful tool to support and verify solutions found through algebraic methods.

Algebraic Manipulation

Algebraic manipulation is an essential skill for solving equations. It involves rearranging the equation to isolate the variable you are solving for. This often means shifting terms across the equation using addition, subtraction, multiplication, or division.
In the given example, we started with \( 2x - 5 = x + 7 \). Our first step was to bring all variable terms to one side. We subtracted \( x \) from both sides to keep the equation balanced, simplifying to \( x - 5 = 7 \).
Next, the goal was isolating \( x \). We did this by adding 5 to both sides, resulting in \( x = 12 \).
These steps demonstrate the process of using algebraic manipulation to solve for an unknown variable. By consistently applying basic arithmetic operations while maintaining the equation's balance, we can arrive at the correct solution.

Verification of Solutions

Verification is a paramount step in solving equations as it assures the solution's accuracy. In essence, this step involves substituting the found solution back into the original equation to check its validity.
For this problem, after determining \( x = 12 \), we substituted this value back into both sides of the original equation to check:

• \( 2(12) - 5 = 12 + 7 \)
• Simplifies to \( 24 - 5 = 19 \)
• Which is \( 19 = 19 \)

Because both sides of the equation are equal, the solution is verified. This step is crucial in ensuring there was no mistake in the algebraic process and that the found value indeed satisfies the equation.