Let's solve the problem step-by-step to determine which equation can be used to find the value of [tex]\( x \)[/tex] for the given isosceles triangle.
1. Identify the given values and what they represent:
- Perimeter of the isosceles triangle: [tex]\( 7.5 \)[/tex] meters.
- Shortest side [tex]\( y \)[/tex]: [tex]\( 2.1 \)[/tex] meters.
2. Understand the properties of an isosceles triangle:
- An isosceles triangle has two sides of equal length, and the third side (base) may be different.
- Let's denote the lengths of the two equal sides as [tex]\( x \)[/tex].
3. Write the perimeter equation:
- The perimeter [tex]\( P \)[/tex] of a triangle is the sum of all its sides.
- In this case: [tex]\( P = x + x + y \)[/tex].
- Substituting the given perimeter and shortest side:
[tex]\[
7.5 = x + x + 2.1
\][/tex]
4. Simplify the equation:
- Combine like terms (the two [tex]\( x \)[/tex] terms):
[tex]\[
7.5 = 2x + 2.1
\][/tex]
5. Rearrange the equation to isolate [tex]\( x \)[/tex]:
- Subtract [tex]\( 2.1 \)[/tex] from both sides to isolate the term involving [tex]\( x \)[/tex]:
[tex]\[
7.5 - 2.1 = 2x
\][/tex]
6. Simplified equation:
- This simplifies to:
[tex]\[
2x = 5.4
\][/tex]
- While this is a crucial step in solving for [tex]\( x \)[/tex], the equation we need is before this step right after substituting given values:
[tex]\[
2.1 + 2x = 7.5
\][/tex]
7. Check the choices given:
- [tex]\( 2x - 2.1 = 7.5 \)[/tex] is not correct since it doesn't correctly reformulate the perimeter equation.
- [tex]\( 4.2 + y = 7.5 \)[/tex] does not involve [tex]\( x \)[/tex] and is irrelevant.
- [tex]\( v - 4.2 = 7.5 \)[/tex] has incorrect symbols and does not align with the given question context.
- [tex]\( 2.1 + 2x = 7.5 \)[/tex] is indeed correct as shown in our steps.
Therefore, the equation to find the value of [tex]\( x \)[/tex] given that the shortest side [tex]\( y \)[/tex] measures [tex]\( 2.1 \)[/tex] meters and the perimeter is [tex]\( 7.5 \)[/tex] meters is:
[tex]\[
2.1 + 2x = 7.5
\][/tex]
