The image presents a geometry problem with line segments labeled as $\overline{BC}$ and $\overline{BD}$ along a straight line. The information provided is as follows:

• $BD = 7x - 10$
• $BC = 4x - 2$
• $CD = x + 5$

The problem asks to find each value.

Approach:

1. The points B, C, and D lie on the same line. This means that the sum of $BC$ and $CD$ equals $BD$. So, we can set up the equation: $BC + CD = BD$ Substituting the expressions for each segment: $(4x - 2) + (x + 5) = 7x - 10$

2. Solve for $x$: Simplifying the left side: $5x + 3 = 7x - 10$ Move the terms involving $x$ to one side: $5x - 7x = -10 - 3$ $-2x = -13$ Divide by -2: $x = \frac{13}{2} = 6.5$

3. Now, substitute $x = 6.5$ back into the expressions for $BD$, $BC$, and $CD$ to find their values:

   • $BD = 7(6.5) - 10 = 45.5 - 10 = 35.5$
   • $BC = 4(6.5) - 2 = 26 - 2 = 24$
   • $CD = 6.5 + 5 = 11.5$

Thus, the values are:

• $BD = 35.5$
• $BC = 24$
• $CD = 11.5$

Would you like more details or further clarification?

Related questions:

1. What happens if we adjust the equation $BD = 7x - 10$ by adding constraints?
2. How can we verify that the points B, C, and D are collinear using these values?
3. Can we use these principles for any other geometric figure or only for line segments?
4. How would the problem change if one of the equations had an additional variable?
5. What happens if the value of $x$ turns out to be negative in similar problems?

Tip:

Always double-check the relations between line segments in geometry problems to avoid missing key constraints like collinearity.