Question

in a small town, there are 4 times as many left-handed males as there are left-handed females, and there are 3 times as many right-handed females as there are right-handed males. there are a total of 204 males and 348 females in the town.

let \\(x\\) represent the number of left-handed females, and let \\(y\\) represent the number of right-handed males. write a system of equations to represent the situation. what is the value of \\(x\\), the number of left-handed females?

6
24
96
108

Explanation:

Identify and define variables

Using the Variable Identification knowledge point

• Let \(x\) be the number of left-handed females.
• Let \(y\) be the number of right-handed males.
• Left-handed males: \(4x\) (since there are 4 times as many left-handed males as left-handed females).
• Right-handed females: \(3y\) (since there are 3 times as many right-handed females as right-handed males).

Translate relationships to equations

Using the Algebraic Translation and System of Equations Translation knowledge points

• Total males: \(\text{Left-handed males} + \text{Right-handed males} = 204\)

$$4x + y = 204$$

• Total females: \(\text{Left-handed females} + \text{Right-handed females} = 348\)

$$x + 3y = 348$$

Solve the system of equations

Using the Substitution Method knowledge point

• From the first equation, express \(y\) in terms of \(x\):

$$y = 204 - 4x$$

• Substitute \(y\) into the second equation:

$$x + 3(204 - 4x) = 348$$

$$x + 612 - 12x = 348$$

$$-11x + 612 = 348$$

$$-11x = -264$$

$$x = 24$$

Verify the solution

Using the Substitution Method knowledge point

• If \(x = 24\):

$$y = 204 - 4(24) = 204 - 96 = 108$$

• Check in the second equation:

$$24 + 3(108) = 24 + 324 = 348$$

• The solution is correct.

Answer:

• (A) 6
• (B) 24 (Correct answer)
• (C) 96
• (D) 108