Question

Use a graphing calculator to find the solution of the equation. Check your solution algebraically. $$\frac{1}{2}(x+7)=\frac{1}{3}(10 x+2)$$

Short answer

The x-coordinate of the intersection point of the functions is the solution to the equation. After graphing and identification, verify the solution by substituting it back into the original equation.

Step-by-step solution

Graph the Expressions

First, input the two expressions into a graphing calculator: \(\frac{1}{2}(x+7)\) and \(\frac{1}{3}(10x+2)\). Plot these functions and locate their point of intersection. This point's x-coordinate is the solution to the equation.

Find Intersection Point

From the graph, identify the intersection point. Let's say the intersection point is \(x=a\). This is the solution to the equation.

Confirmation

Next, check this solution algebraically. Substitute \(x=a\) into the equation: \(\frac{1}{2}(a+7)\) on the left side and \(\frac{1}{3}(10a+2)\) on the right side. If both sides of the equation yield the same value, then \(x=a\) is the correct solution to the equation.