Question

from unit 4, lesson 4
consider the linear equation \\(9x - 3y = 12\\).
a. the pair \\((3, 5)\\) is a solution to the equation. find another pair \\((x, y)\\) that is a solution to the equation.
b. are \\((3, 5)\\) and \\((2, -10)\\) solutions to the inequality \\(9x - 3y \le 12\\)?
explain how you know.

Explanation:

Find another solution to the equation

We choose an arbitrary value for \(x\) to find a corresponding \(y\). Let \(x = 2\).
Using the Linear Inequality Solutions knowledge point

$$
LATEXBLOCK0
$$

Thus, another solution is \((2, 2)\).

Test the first point in the inequality

We substitute the point \((3, 5)\) into the inequality \(9x - 3y \le 12\).
Using the Linear Inequality Solutions knowledge point

$$
LATEXBLOCK1
$$

Since \(12 \le 12\) is true, \((3, 5)\) is a solution.

Test the second point in the inequality

We substitute the point \((2, -10)\) into the inequality \(9x - 3y \le 12\).
Using the Linear Inequality Solutions knowledge point

$$
LATEXBLOCK2
$$

Since \(48 \le 12\) is false, \((2, -10)\) is not a solution.

Answer:

Question a

Another solution is \((2, 2)\).

Question b

Yes, \((3, 5)\) is a solution because substituting the values gives \(12 \le 12\), which is true. No, \((2, -10)\) is not a solution because substituting the values gives \(48 \le 12\), which is false.