Problem 81 - College Algebra with Modeling and Visualization

Question

Solve the equation (to the nearest tenth) (a) symbolically, (b) graphically, and (c) numerically. $$ 6 x-8=-7 x+18 $$

Step-by-Step Solution

Answer

The solution is \(x = 2\).

Step 1: Simplify the Equation

First, combine like terms. We start with the equation \(6x - 8 = -7x + 18\). Add \(7x\) to both sides to get all \(x\) terms on one side: \(6x + 7x - 8 = 18\). This simplifies to \(13x - 8 = 18\).

Step 2: Solve for x Symbolically

Now, solve for \(x\) by isolating it. Add 8 to both sides: \(13x = 26\). Then, divide by 13 to solve for \(x\): \(x = 2\).

Step 3: Verify Solution Graphically

Plot the equations \(y = 6x - 8\) and \(y = -7x + 18\) on a graph. The solution is where these two lines intersect, which should be at \(x = 2\).

Step 4: Verify Solution Numerically

Substitute \(x = 2\) back into the original equation to verify: Left side \(= 6(2) - 8 = 4\), Right side \(= -7(2) + 18 = 4\). Both sides equal, confirming \(x = 2\) is correct.

Key Concepts

Symbolic Solution

When solving linear equations symbolically, we manipulate the equation using algebraic techniques until we isolate the variable on one side. This process involves step-by-step simplification and transformation of the equation. Let's take the linear equation given in the exercise:
\[6x - 8 = -7x + 18\]

• First, we combine the like terms by adding \(7x\) to both sides. This moves all terms containing \(x\) to one side of the equation, leading to \(13x - 8 = 18\).
• Next, we aim to isolate \(x\). To do this, add \(8\) to both sides of the equation so that it reads \(13x = 26\).
• Finally, divide both sides by \(13\) to solve for \(x\), giving \(x = 2\).

The symbolic solution using these steps provides us with the value of \(x\), which is 2. Symbolic solutions are precise and involve using rules of arithmetic and algebra to simplify and solve equations.

Graphical Solution

A graphical solution involves plotting the equations on a coordinate plane and finding the intersection point of the lines. In the context of this exercise, we graph the equations \(y = 6x - 8\) and \(y = -7x + 18\).

• The first step is to plot the line \(y = 6x - 8\). This line will have a slope of 6 and will intersect the y-axis at \(y = -8\).
• The second step is to plot the line \(y = -7x + 18\), which has a slope of -7 and a y-intercept of \(18\).
• After plotting, you will observe that the two lines intersect at a single point. This intersection represents the solution to the equation.

In this case, the lines intersect at \(x = 2\). The graphical solution offers a visual representation of solving the equation and confirms that the value obtained symbolically is accurate.

Numerical Verification

Numerical verification is the process of checking if the solution to an equation is correct by substituting it back into the original equation. Using this method, we assess whether both sides of the equation produce the same numerical result.
For the exercise equation \(6x - 8 = -7x + 18\) with \(x = 2\):

• Substitute \(x = 2\) into the left-hand side: Compute \(6(2) - 8\), which equals \(12 - 8 = 4\).
• Next, substitute \(x = 2\) into the right-hand side: Compute \(-7(2) + 18\), which equals \(-14 + 18 = 4\).
• Both sides equal 4, confirming that the solution \(x = 2\) is indeed correct.

Numerical verification is a practical way to affirm the symbolic and graphical solutions, ensuring consistency and accuracy in solving the equation.