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The perimeters of the two figures have the same measure. Write and solve an algebraic equation to find the value of 
x. 

Let's analyze the given shapes and write the algebraic equation.

### For the Rectangle:
- The length is \( 5x - 2 \).
- The width is \( 3x \).

The perimeter \( P \) of a rectangle is given by:
\[
P_{\text{rectangle}} = 2(\text{length} + \text{width})
\]
Substituting the given values:
\[
P_{\text{rectangle}} = 2((5x - 2) + 3x)
\]
Simplifying:
\[
P_{\text{rectangle}} = 2(8x - 2) = 16x - 4
\]

### For the Triangle:
- The sides are \( x + 10 \), \( 2x + 3 \), and \( 10x - 5 \).

The perimeter \( P \) of a triangle is the sum of the lengths of its sides:
\[
P_{\text{triangle}} = (x + 10) + (2x + 3) + (10x - 5)
\]
Simplifying:
\[
P_{\text{triangle}} = x + 10 + 2x + 3 + 10x - 5 = 13x + 8
\]

### Set the Perimeters Equal:
Since the perimeters of the two shapes are the same:
\[
16x - 4 = 13x + 8
\]

### Solve for \( x \):
1. Subtract \( 13x \) from both sides:
\[
16x - 13x - 4 = 8
\]
\[
3x - 4 = 8
\]

2. Add 4 to both sides:
\[
3x = 12
\]

3. Divide by 3:
\[
x = 4
\]

Thus, the value of \( x \) is \( \boxed{4} \).

Let me know if you would like further details or clarifications.

### Related Questions:
1. What is the perimeter of the rectangle when \( x = 4 \)?
2. What is the perimeter of the triangle when \( x = 4 \)?
3. Can you find the area of the rectangle with the given dimensions?
4. What are the lengths of the triangle's sides when \( x = 4 \)?
5. How would the equation change if the perimeter of the triangle was greater than the rectangle?

### Tip:
Always double-check the perimeters of both shapes after solving to ensure accuracy.

The perimeters of the two figures have the same measure. Write and solve an algebraic equation to find the value of x.

Learn how to find the value of x by setting the perimeters of a rectangle and triangle equal. This problem covers key algebraic concepts and geometric perimeter formulas. Suitable for students in Grades 7-9.

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