Solve each problem by forming and then solving a pair of simultaneous equations: At a meeting of the members of a certain club, a proposal was voted on. If 357 members voted and

the proposal was carried by a majority of 21, how many voted for and how many voted against?

To solve this problem, let's define the variables and set up a system of simultaneous equations:

Let:
- \( x \) = number of members who voted for the proposal.
- \( y \) = number of members who voted against the proposal.

We are given two key pieces of information:
1. The total number of members who voted is 357.
2. The proposal was carried by a majority of 21 votes.

From the first piece of information, we can form the equation:

\[
x + y = 357 \quad \text{(Equation 1)}
\]

From the second piece of information, we know that the number of votes for the proposal exceeded the number of votes against by 21:

\[
x = y + 21 \quad \text{(Equation 2)}
\]

Now, let's solve these equations simultaneously.

**Substituting Equation 2 into Equation 1:**

\[
(y + 21) + y = 357
\]

Simplify and solve for \( y \):

\[
2y + 21 = 357
\]

\[
2y = 357 - 21
\]

\[
2y = 336
\]

\[
y = 168
\]

Now, substitute the value of \( y \) back into Equation 2 to find \( x \):

\[
x = 168 + 21
\]

\[
x = 189
\]

**Solution:**
- The number of members who voted for the proposal is \( 189 \).
- The number of members who voted against the proposal is \( 168 \).

Would you like further details or have any questions?

Here are some related questions:
1. What if the total number of members who voted was different?
2. How would the solution change if the majority was different?
3. What if we knew the ratio of those who voted for to against instead of the majority?
4. How do simultaneous equations help in real-world decision-making scenarios?
5. Can these equations be solved using matrix methods?

**Tip:** Always check the equations you form by substituting the solutions back into the original equations to verify your answer!

Learn how to solve a problem involving simultaneous equations by determining the number of votes for and against a proposal. This problem is solved using algebraic methods, specifically simultaneous equations. Suitable for students in Grades 9-12.

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