Question

1. fill in the blanks using the available answer choices.

jason graphed the system of equations, but did not shade the regions in a way that helps him find the solution.
( y < x - 1 )
( -x + 2y leq 1 )

in which region of the graph is the solution to this system found?
section __________
(blank 1)
blank 1 options

• i
• ii
• iii
• iv

1. solve the system of equations using elimination.

( 3x - 3y = -3 )
( - 3y - 3x = 9 )
the system has no solution.
( (-2, -1) )
( (-1, 0) )
( (-3, 0) )
this system has infinite solutions.

Explanation:

Question 16

Step 1: Analyze \( y < x - 1 \)

For the inequality \( y < x - 1 \), the boundary line is dashed (since it's \( < \)) and we shade below the line \( y = x - 1 \).

Step 2: Analyze \( -x + 2y \leq 1 \)

Rewrite it as \( 2y \leq x + 1 \) or \( y \leq \frac{1}{2}x + \frac{1}{2} \). The boundary line is solid, and we shade below this line.

Step 3: Find the overlapping region

We test a point in each section. Let's take the origin \((0,0)\) for \( y < x - 1 \): \( 0 < 0 - 1 \) is false. For \( -x + 2y \leq 1 \): \( 0 + 0 \leq 1 \) is true. Now check section II (let's pick a point like \((3,1)\)): For \( y < x - 1 \): \( 1 < 3 - 1 = 2 \) (true). For \( -x + 2y \leq 1 \): \( -3 + 2(1) = -1 \leq 1 \) (true). So the overlapping region (solution) is in section II.

Step 1: Rewrite the equations

The system is:
\[

$$\begin{cases} 3x - 3y = -3 \\ -3x - 3y = 9 \end{cases}$$

\]

Step 2: Add the two equations

Add the left sides and right sides: \((3x - 3y) + (-3x - 3y) = -3 + 9\)
Simplify: \( -6y = 6 \)

Step 3: Solve for \( y \)

Divide both sides by \(-6\): \( y = \frac{6}{-6} = -1 \)

Step 4: Substitute \( y = -1 \) into first equation

\( 3x - 3(-1) = -3 \)
Simplify: \( 3x + 3 = -3 \)
Subtract 3: \( 3x = -6 \)
Divide by 3: \( x = -2 \)
So the solution is \((-2, -1)\)

Answer:

II

Question 17