Let's determine which equation is false based on the given solution set [tex]\(S: \{2\}\)[/tex]. The value provided for the variables [tex]\(p\)[/tex], [tex]\(t\)[/tex], [tex]\(c\)[/tex], and [tex]\(m\)[/tex] is 2.
1. Equation 1: [tex]\(7 = 6p - 5\)[/tex]
- Substitute [tex]\(p = 2\)[/tex].
- [tex]\(6p - 5 = 6(2) - 5 = 12 - 5 = 7\)[/tex]
- Therefore, [tex]\(7 = 7\)[/tex], which is true.
2. Equation 2: [tex]\(4t = 8\)[/tex]
- Substitute [tex]\(t = 2\)[/tex].
- [tex]\(4t = 4(2) = 8\)[/tex]
- Therefore, [tex]\(8 = 8\)[/tex], which is true.
3. Equation 3: [tex]\(3(3c + 1) = 21\)[/tex]
- Substitute [tex]\(c = 2\)[/tex].
- [tex]\(3(3c + 1) = 3(3(2) + 1) = 3(6 + 1) = 3(7) = 21\)[/tex]
- Therefore, [tex]\(21 = 21\)[/tex], which is true.
4. Equation 4: [tex]\(5m + 8 = 16\)[/tex]
- Substitute [tex]\(m = 2\)[/tex].
- [tex]\(5m + 8 = 5(2) + 8 = 10 + 8 = 18\)[/tex]
- Therefore, [tex]\(18 \neq 16\)[/tex], which is false.
From the steps above, we observe that the equations:
- [tex]\(7 = 6p - 5\)[/tex]
- [tex]\(4t = 8\)[/tex]
- [tex]\(3(3c + 1) = 21\)[/tex]
are all true when substituting the value 2 into the variable. The equation:
- [tex]\(5m + 8 = 16\)[/tex]
is false when substituting the value 2 into [tex]\(m\)[/tex]. Hence, the false equation is:
[tex]\[5m + 8 = 16\][/tex]
