The question asks for the solution (x, y) to the system (1/2)x + (1/3)y = 4 and x - (1/3)y = 2. I will use elimination since y coefficients are opposites. Adding the equations: (1/2)x + x + (1/3)y - (1/3)y = 4 + 2, (3/2)x = 6, x = 6*(2/3) = 4. Substitute into second: 4 - (1/3)y = 2, -(1/3)y = -2, y = 6. Verify in first: (1/2)*4 + (1/3)*6 = 2 + 2 = 4, correct. A key error is mishandling fractions during addition. Clear fractions first by multiplying equations if it simplifies steps.

The question asks how many solutions the system of Line 1 through (-2,1) and (2,9), and Line 2 through (0,5) and (4,13) has. First, find slopes: Line 1 m = (9-1)/(2+2) = 8/4 = 2; Line 2 m = (13-5)/(4-0) = 8/4 = 2, same slope. Equation for Line 1: y - 1 = 2(x + 2), y = 2x + 5; Line 2: y - 5 = 2(x - 0), y = 2x + 5, same line. Thus, they coincide with infinitely many solutions. A common error is miscalculating slope or y-intercept. When slopes match, check if y-intercepts are equal to confirm dependence.

The question asks for the number of floor tickets y sold, given the system x + y = 140 and 12x + 20y = 2320, where x is balcony tickets. I will use substitution: from first, x = 140 - y; plug into second: 12(140 - y) + 20y = 2320, 1680 - 12y + 20y = 2320, 1680 + 8y = 2320, 8y = 640, y = 80. Verify: x = 60, 1260 + 2080 = 720 + 1600 = 2320, correct. A key error is misapplying distribution in substitution. In ticket problems, ensure the solution yields non-negative integers that fit the context.

The question asks which ordered pair (x, y) satisfies both 4x + y = 9 and 2x - y = 3. I will use the elimination method as the y terms are opposites. Adding the equations: 4x + 2x + y - y = 9 + 3, so 6x = 12, x = 2; then substitute into first: 42 + y = 9, 8 + y = 9, y = 1, so (2,1). Verify in second: 22 - 1 = 4 - 1 = 3, correct. A common error is subtracting instead of adding, leading to wrong elimination. A good strategy is to plug all choices into both equations to quickly identify the correct one.

Multiply the first equation by 3 and the second by 2: $6x+3y=27$ and $-6x+4y=-6$; adding gives $7y=21\Rightarrow y=3$, then $x=3$. The other options result from arithmetic or sign errors when combining equations.

Add the equations to eliminate $y$: $(x+y)+(2x-y)=11+7\Rightarrow 3x=18\Rightarrow x=6$, then $y=11-6=5$. The distractors come from swapping $x$ and $y$ or miscomputing during elimination.

Multiply the first equation by 2 to get $x+2y=14$, then subtract $(x-y=5)$ to obtain $3y=9\Rightarrow y=3$ and $x=8$. Other choices reflect swapping coordinates, sign errors, or coefficient arithmetic slips.

Use $x=7-y$ in $3x-2y=1$: $3(7-y)-2y=1\Rightarrow 21-3y-2y=1\Rightarrow 5y=20\Rightarrow y=4$ and then $x=3$. Distractors come from swapping $x$ and $y$ or arithmetic mistakes when combining terms.

From $x-y=1$, substitute $x=y+1$ into $2x+3y=17$ to get $2(y+1)+3y=17\Rightarrow 5y=15\Rightarrow y=3$, then $x=4$. The other choices reflect swapped coordinates, sign errors, or arithmetic slips.

From $2(x + y) = 14$, simplify to $x + y = 7$; adding with $3x - y = 5$ gives $4x = 12$, so $x = 3$ and $y = 4$. Other choices either swap the values or satisfy only one of the equations.