To solve for the value of \( x \) in the given isosceles triangle with a perimeter of \( 7.5 \) meters, where the shortest side \( y \) measures \( 2.1 \) meters, we need to find the correct equation among the given choices.
First, let's recall that in an isosceles triangle, two sides are equal and one side is different. Given the perimeter and the length of the shortest side, we need to form an equation using this information.
### Step-by-Step Solution:
1. Understanding the Perimeter:
- The perimeter of the triangle is the sum of all its sides.
- The triangle has a perimeter \( P \) of \( 7.5 \) meters.
- One side is given as \( y = 2.1 \) meters.
2. Isosceles Triangle Property:
- Since it's an isosceles triangle, let the two equal sides be each \( x \).
- The shortest side, \( y \), is \( 2.1 \) meters.
- Therefore, the two equal sides, both \( x \), along with the shortest side, sum to the perimeter.
- So, we have: \( x + x + y = 7.5 \).
3. Form the Equation:
- Substitute \( y = 2.1 \) into the perimeter equation:
[tex]\[
x + x + 2.1 = 7.5
\][/tex]
- Combine like terms:
[tex]\[
2x + 2.1 = 7.5
\][/tex]
4. Identify the correct equation among the given choices:
- Now we need to match this equation with one of the given options:
1. \( 2x - 2.1 = 7.5 \): This does not match.
2. \( 4.2 + y = 7.5 \): Substituting \( y = 2.1 \) into this would give \( 4.2 + 2.1 = 6.3 \neq 7.5 \).
3. \( y - 4.2 = 7.5 \): Substituting \( y = 2.1 \) into this would give \( 2.1 - 4.2 = -2.1 \neq 7.5 \).
4. \( 2.1 + 2x = 7.5 \): This matches our derived equation \( 2x + 2.1 = 7.5 \).
Thus, the equation that can be used to find the value of \( x \) is:
[tex]\[
\boxed{2.1 + 2x = 7.5}
\][/tex]
