Answer:
20. x = 5, y = -2
21. x = 11, y = 12
22. x = 22, y = 11
23. x = 11, y = 10
Step-by-step explanation:
20. Opposite sides in the parallelogram are congruent, so
[tex]2y+18=3x-1\\ \\6x-3=17-5y[/tex]
Solve this system of two equations:
[tex]\left\{\begin{array}{l}2y-3x=-19\\ \\6x+5y=20\end{array}\right.[/tex]
Multiply the first equation by 2 and add two equations:
[tex]2(2y-3x)+6x+5y=2\cdot (-19)+20\\ \\4y-6x+6x+5y=-38+20\\ \\9y=-18\\ \\y=-2[/tex]
Substitute it into the first equation:
[tex]2\cdot (-2)-3x=-19\\ \\-3x=-19+4\\ \\-3x=-15\\ \\x=5[/tex]
21. Opposite angles in the parallelogram are congruent, so
[tex]11x+5=10y+6[/tex]
Consecutive angles are supplementary, so
[tex]6x-y+11x+5=180^{\circ}[/tex]
Solve this system of two equations:
[tex]\left\{\begin{array}{l}11x-10y=1\\ \\17x-y=175\end{array}\right.[/tex]
From the second equation
[tex]y=17x-175[/tex]
Substitute it into the first equation:
[tex]11x-10(17x-175)=1\\ \\11x-170x+1750=1\\ \\-159x=-1749\\ \\x=11\\ \\y=17\cdot 11-175=187-175=12[/tex]
22. Opposite angles in the parallelogram are congruent, so
[tex]2x-5=3y-12[/tex]
Consecutive angles are supplementary, so
[tex]2x-5+7y+x=180^{\circ}[/tex]
Solve this system of two equations:
[tex]\left\{\begin{array}{l}2x-3y=-7\\ \\3x+7y=185\end{array}\right.[/tex]
From the first equation
[tex]x=-3.5+1.5y[/tex]
Substitute it into the second equation:
[tex]3(-3.5+1.5y)+7y=185\\ \\-10.5+4.5y+7y=185\\ \\-105+45y+70y=1,850\\ \\115y=1,850+105\\ \\115y=1,955\\ \\y=17\\ \\x=-3.5+1.5\cdot 17=22[/tex]
23. Opposite sides in the parallelogram are congruent, so
[tex]2x+9=4y-9\\ \\3x-5=2y+8[/tex]
Solve this system of two equations:
[tex]\left\{\begin{array}{l}2x-4y=-18\\ \\3x-2y=13\end{array}\right.[/tex]
Multiply the second equation by 2 and subtract it from the first equation:
[tex]2x-4y-2(3x-2y)=-18-2\cdot 13\\ \\2x-4y-6x+4y=-18-26\\ \\-4x=-44\\ \\x=11[/tex]
Substitute it into the first equation:
[tex]2\cdot 11-4y=-18\\ \\-4y=-18-22\\ \\-4y=-40\\ \\y=10[/tex]
