Let's break down the solution step-by-step for the equation [tex]\(21 + 2w = 62\)[/tex], given that Shana wants to make the length of the run 20 feet.
1. We start with the equation:
[tex]\[
21 + 2w = 62
\][/tex]
2. The first step is to isolate the term with the variable [tex]\(w\)[/tex]. To do this, we need to subtract 21 from both sides of the equation:
[tex]\[
2w = 62 - 21
\][/tex]
3. Simplifying the right-hand side, we get:
[tex]\[
2w = 41
\][/tex]
4. Next, we divide both sides by 2 to solve for [tex]\(w\)[/tex]:
[tex]\[
w = \frac{41}{2}
\][/tex]
5. This simplifies to:
[tex]\[
w = 20.5
\][/tex]
Now let's address the given statements:
- The value of [tex]\(w\)[/tex] is 10 feet.
- This statement is false because our calculated value of [tex]\(w\)[/tex] is 20.5 feet, not 10 feet.
- The value of [tex]\(w\)[/tex] can be zero.
- This statement is true. Although in this specific problem [tex]\(w\)[/tex] is not zero, [tex]\(w\)[/tex] could theoretically be zero because a width of zero is possible in general mathematical realms, even if impractical in this scenario.
- The value of [tex]\(w\)[/tex] cannot be a negative number.
- This statement is true because dimensions of a physical object such as a dog run cannot be negative in the real world context.
- Substitution is used to replace the variable [tex]\(l\)[/tex] with a value of 20.
- This statement is true. We substituted [tex]\(l\)[/tex] with 20 and subsequently solved for [tex]\(w\)[/tex].
- The subtraction property of equality is used to isolate the term with the variable [tex]\(w\)[/tex].
- This statement is true. By subtracting 21 from both sides, we isolated the term containing [tex]\(w\)[/tex].
Based on this detailed analysis, the statements that are true of the solution are:
- The value of [tex]\(w\)[/tex] can be zero.
- The value of [tex]\(w\)[/tex] cannot be a negative number.
- Substitution is used to replace the variable [tex]\(l\)[/tex] with a value of 20.
- The subtraction property of equality is used to isolate the term with the variable [tex]\(w\)[/tex].
