Question

the difference between twice a number, \\(x\\), and a smaller number, \\(y\\), is 3. the sum of twice the number and the smaller number is \\(-3\\). which equations represent this situation?

\\(y = 2x - 3\\) and \\(y = -2x - 3\\)

\\(y = 2x - 3\\) and \\(y = -\frac{1}{2}x - \frac{3}{2}\\)

\\(y = 2x + 3\\) and \\(y = -2x - 3\\)

\\(y = 2x + 3\\) and \\(y = -\frac{1}{2}x - \frac{3}{2}\\)

Explanation:

Identify variables and translate the first statement

Using the Variable Identification and Algebraic Translation knowledge points
The problem defines:

• Larger number: \(x\)
• Smaller number: \(y\)

The first statement is: "The difference between twice a number, \(x\), and a smaller number, \(y\), is 3."

$$ 2x - y = 3 $$

Translate the second statement

Using the Algebraic Translation knowledge point
The second statement is: "The sum of twice the number and the smaller number is \(-3\)."

$$ 2x + y = -3 $$

Convert the first equation to slope-intercept form

Using the Slope-Intercept Form knowledge point

$$
LATEXBLOCK0
$$

Convert the second equation to slope-intercept form

Using the Slope-Intercept Form knowledge point

$$
LATEXBLOCK1
$$

Match with the given options

Using the Slope-Intercept Form knowledge point
The system of equations in slope-intercept form is:

$$ y = 2x - 3 \quad \text{and} \quad y = -2x - 3 $$

This matches the first option.

Answer:

• (A) \(y = 2x - 3\) and \(y = -2x - 3\) (Correct answer)
• (B) \(y = 2x - 3\) and \(y = -\frac{1}{2}x - \frac{3}{2}\)
• (C) \(y = 2x + 3\) and \(y = -2x - 3\)
• (D) \(y = 2x + 3\) and \(y = -\frac{1}{2}x - \frac{3}{2}\)