High School

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Identify the equations below that are equivalent to:

[tex]3 = 17c[/tex]

\[
\begin{array}{ll}
11 = 17c + 8 & 7 = 17c + 4 \\
16 = 13 + 17c & 19 = 17 + 17c
\end{array}
\]

Answer :

To determine which of the given equations are equivalent to the original equation [tex]\(3 = 17c\)[/tex], we need to solve each of the given equations for [tex]\(c\)[/tex] and see if any result in the same solution for [tex]\(c\)[/tex] as the original equation.

Let's solve each equation one-by-one:

1. Original Equation:
[tex]\[
3 = 17c
\][/tex]
Solving for [tex]\(c\)[/tex], we divide both sides by 17:
[tex]\[
c = \frac{3}{17}
\][/tex]

2. First Given Equation:
[tex]\[
11 = 17c + 8
\][/tex]
To solve for [tex]\(c\)[/tex], first subtract 8 from both sides:
[tex]\[
3 = 17c
\][/tex]
Divide both sides by 17:
[tex]\[
c = \frac{3}{17}
\][/tex]

3. Second Given Equation:
[tex]\[
7 = 17c + 4
\][/tex]
To solve for [tex]\(c\)[/tex], first subtract 4 from both sides:
[tex]\[
3 = 17c
\][/tex]
Divide both sides by 17:
[tex]\[
c = \frac{3}{17}
\][/tex]

4. Third Given Equation:
[tex]\[
16 = 13 + 17c
\][/tex]
To solve for [tex]\(c\)[/tex], first subtract 13 from both sides:
[tex]\[
3 = 17c
\][/tex]
Divide both sides by 17:
[tex]\[
c = \frac{3}{17}
\][/tex]

5. Fourth Given Equation:
[tex]\[
19 = 17 + 17c
\][/tex]
To solve for [tex]\(c\)[/tex], first subtract 17 from both sides:
[tex]\[
2 = 17c
\][/tex]
Divide both sides by 17:
[tex]\[
c = \frac{2}{17}
\][/tex]

After solving all the equations, we see that only the fourth equation [tex]\(19 = 17 + 17c\)[/tex] leads to a different value for [tex]\(c\)[/tex]. Therefore, none of the altered equations are equivalent to the original equation [tex]\(3 = 17c\)[/tex] in terms of having the same solution for [tex]\(c\)[/tex]. Hence, there are no equations equivalent to [tex]\(3 = 17c\)[/tex] within the given options.

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