Answer Key 5.3

$\begin{array}{rrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &4x&+&2y&=&0 \\ +&-4x&-&9y&=&-28 \\ \midrule &&&\dfrac{-7y}{-7}&=&\dfrac{-28}{-7} \\ \\ &&&y&=&4 \\ \\ &4x&+&2(4)&=&0 \\ &4x&+&8&=&0 \\ &&-&8&&-8 \\ \midrule &&&\dfrac{4x}{4}&=&\dfrac{-8}{4} \\ \\ &&&x&=&-2 \\ (-2,4)&&&&& \end{array}$
$\begin{array}{rrrcrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &-7x&+&y&=&-10 \\ +&-9x&-&y&=&-22 \\ \midrule &&&\dfrac{-16x}{-16}&=&\dfrac{-32}{-16} \\ \\ &&&x&=&2 \\ \\ &-7(2)&+&y&=&-10 \\ &-14&+&y&=&-10 \\ &+14&&&&+14 \\ \midrule &&&y&=&4 \\ (2,4)&&&&& \end{array}$
$\begin{array}{rrrrrr} \\ \\ &-9x&+&5y&=&-22 \\ +&9x&-&5y&=&13 \\ \midrule &&&0&=&-9 \end{array}$ ∴ $\text{Two parallel lines. No solution}$
$\begin{array}{rrrrrr} \\ \\ &-x&-&2y&=&-7 \\ +&x&+&2y&=&7 \\ \midrule &&&0&=&0 \end{array}$ ∴ $\text{Two identical lines. Infinite solutions}$
$\begin{array}{rrrrrr} \\ \\ &-6x&+&9y&=&3 \\ +&6x&-&9y&=&-9 \\ \midrule &&&0&=&-6 \end{array}$ ∴ $\text{Two parallel lines. No solution}$
$\begin{array}{rrrrrrr} \\ \\ \\ \\ \\ &5x&-&5y&=&-15&(\div 5) \\ &(x&-&y&=&-3)&(-1) \\ \\ &x&-&y&=&-3& \\ +&-x&+&y&=&3& \\ \midrule &&&0&=&0& \\ \end{array}$ ∴ $\text{Two identical lines. Infinite solutions}$
$\begin{array}{rrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &4x&-&6y&=&-10 \\ +&4x&+&6y&=&-14 \\ \midrule &&&\dfrac{8x}{8}&=&\dfrac{-24}{8} \\ \\ &&&x&=&-3 \\ \\ &4(-3)&-&6y&=&-10 \\ &-12&-&6y&=&-10 \\ &+12&&&&+12 \\ \midrule &&&\dfrac{-6y}{-6}&=&\dfrac{2}{-6} \\ \\ &&&y&=&-\dfrac{1}{3} \\ (-3, -\dfrac{1}{3})&&&&& \end{array}$
$\begin{array}{rrrrrrrl} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &-3x&+&3y&=&-12&\div &(-3) \\ &-3x&+&9y&=&-24&\div &(3) \\ \\ & x&-&y&=&4&& \\ +&-x&+&3y&=&-8&& \\ \midrule &&&\dfrac{2y}{2}&=&\dfrac{-4}{2}&& \\ \\ &&&y&=&-2&& \\ \\ &\therefore x&-&y&=&4&& \\ &x&-&-2&=&4&& \\ &x&+&2&=&4&& \\ &&-&2&&-2&& \\ \midrule &&&x&=&2&& \\ (2,-2)&&&&&&& \end{array}$
$\begin{array}{rrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &(-x&-&5y&=&28)&(-1) \\ \\ &x&+&5y&=&-28& \\ +&-x&+&4y&=&-17& \\ \midrule &&&\dfrac{9y}{9}&=&\dfrac{-45}{9}& \\ \\ &&&y&=&-5& \\ \\ &x&+&5(-5)&=&-28& \\ &x&-&25&=&-28& \\ &&+&25&&+25& \\ \midrule &&&x&=&-3& \\ (-3,-5)&&&&&& \end{array}$
$\begin{array}{rrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &(-10x&-&5y&=&0)&(-1) \\ \\ &10x&+&5y&=&0& \\ +&-10x&-&10y&=&-30& \\ \midrule &&&\dfrac{-5y}{-5}&=&\dfrac{-30}{-5}& \\ \\ &&&y&=&6& \\ \\ &10x&+&5(6)&=&0& \\ &10x&+&30&=&0& \\ &&-&30&&-30& \\ \midrule &&&\dfrac{10x}{10}&=&\dfrac{-30}{10}& \\ \\ &&&x&=&-3& \\ (-3,6)&&&&&& \end{array}$
$\begin{array}{rrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &(2x&-&y&=&5)&(2) \\ \\ &4x&-&2y&=&10& \\ +&5x&+&2y&=&-28& \\ \midrule &&&\dfrac{9x}{9}&=&\dfrac{-18}{9}& \\ \\ &&&x&=&-2& \\ \\ &2(x)&-&y&=&5& \\ &2(-2)&-&y&=&5& \\ &-4&-&y&=&5& \\ &+4&&&&+4& \\ \midrule &&&-y&=&9& \\ \\ &&&y&=&-9& \\ (-2,-9)&&&&&& \end{array}$
$\begin{array}{rrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &-5x&+&6y&=&-17& \\ &(x&-&2y&=&5)&(3) \\ \\ &-5x&+&6y&=&-17& \\ +&3x&-&6y&=&15& \\ \midrule &&&\dfrac{-2x}{-2}&=&\dfrac{-2}{-2}& \\ \\ &&&x&=&1& \\ \\ &x&-&2y&=&5& \\ &1&-&2y&=&5& \\ &-1&&&&-1& \\ \midrule &&&\dfrac{-2y}{-2}&=&\dfrac{4}{-2}& \\ \\ &&&y&=&-2& \\ (1,-2)&&&&&& \end{array}$
$\begin{array}{rrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &(10x&+&6y&=&24)&(\div 2) \\ &(-6x&+&y&=&4)&(-3) \\ \\ &5x&+&3y&=&12& \\ +&18x&-&3y&=&-12& \\ \midrule &&&23x&=&0& \\ &&&x&=&0& \\ \\ &-6(x)&+&y&=&4& \\ &-6(0)&+&y&=&4& \\ &&&y&=&4& \\ (0,4)&&&&&& \end{array}$
$\begin{array}{rrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &(10x&+&6y&=&-10)&(\div -2) \\ \\ &x&+&3y&=&-1& \\ +&-5x&-&3y&=&5& \\ \midrule &&&\dfrac{-4x}{-4}&=&\dfrac{4}{-4}& \\ \\ &&&x&=&-1& \\ \\ &-1&+&3y&=&-1& \\ &+1&&&&+1& \\ \midrule &&&3y&=&0& \\ \\ &&&y&=&0& \\ (-1,0)&&&&&& \end{array}$
$\begin{array}{rrrrrrl} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &(2x&+&4y&=&24)&(\div 2) \\ &(4x&-&12y&=&8)&(\div -4) \\ \\ &x&+&2y&=&12& \\ +&-x&+&3y&=&-2& \\ \midrule &&&\dfrac{5y}{5}&=&\dfrac{10}{5}& \\ \\ &&&y&=&2& \\ \\ &x&+&2(y)&=&12& \\ &x&+&2(2)&=&12& \\ &x&+&4&=&12& \\ &&-&4&&-4& \\ \midrule &&&x&=&8& \\ (8,2)&&&&&& \end{array}$
$\begin{array}{rrrrrrl} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &(-6x&+&4y&=&12)&(\div 2) \\ &(12x&+&6y&=&18)&(\div -3) \\ \\ &-3x&+&2y&=&6& \\ +&-4x&-&2y&=&-6& \\ \midrule &&&-7x&=&0& \\ &&&x&=&0& \\ \\ &\dfrac{-3(x)}{2}&+&\dfrac{2y}{2}&=&\dfrac{6}{2}& \\ \\ &\dfrac{-3(0)}{2}&+&\dfrac{2y}{2}&=&\dfrac{6}{2}& \\ \\ &&&y&=&3& \\ (0,3)&&&&&& \end{array}$
$\begin{array}{rrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &(10x&-&8y&=&-8)&(\div 2) \\ \\ &-7x&+&4y&=&-4& \\ +&5x&-&4y&=&-4& \\ \midrule &&&\dfrac{-2x}{-2}&=&\dfrac{-8}{-2}& \\ \\ &&&x&=&4& \\ \\ &5(4)&-&4y&=&-4& \\ &20&-&4y&=&-4& \\ &-20&&&&-20& \\ \midrule &&&\dfrac{-4y}{-4}&=&\dfrac{-24}{-4}& \\ \\ &&&y&=&6& \\ (4,6)&&&&&& \end{array}$
$\begin{array}{rrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &(-6x&+&4y&=&4)&(\div 2) \\ \\ &-3x&+&2y&=&2& \\ +&3x&-&y&=&26& \\ \midrule &&&y&=&28& \\ \\ &3x&-&28&=&26& \\ &&+&28&&+28& \\ \midrule &&&\dfrac{3x}{3}&=&\dfrac{54}{3}& \\ \\ &&&x&=&18& \\ (18,28)&&&&&& \end{array}$
$\begin{array}{rrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &(-6x&-&5y&=&-3)&(2) \\ \\ &5x&+&10y&=&20& \\ +&-12x&-&10y&=&-6& \\ \midrule &&&\dfrac{-7x}{-7}&=&\dfrac{14}{-7}& \\ \\ &&&x&=&-2& \\ \\ &5(-2)&+&10y&=&20& \\ &-10&+&10y&=&20& \\ &+10&&&&+10& \\ \midrule &&&\dfrac{10y}{10}&=&\dfrac{30}{10}& \\ \\ &&&y&=&3& \\ (-2,3)&&&&&& \end{array}$
$\begin{array}{rrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &(3x&-&7y&=&-11)&(3) \\ \\ &-9x&-&5y&=&-19& \\ +&9x&-&21y&=&-33& \\ \midrule &&&\dfrac{-26y}{-26}&=&\dfrac{-52}{-26}& \\ \\ &&&y&=&2& \\ \\ &3x&-&7(2)&=&-11& \\ &3x&-&14&=&-11& \\ &&+&14&&+14& \\ \midrule &&&\dfrac{3x}{3}&=&\dfrac{3}{3}& \\ \\ &&&x&=&1& \\ (1,2)&&&&&& \end{array}$
$\begin{array}{rrrrrrl} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &(-7x&+&5y&=&-8)&(3) \\ &(-3x&-&3y&=&12)&(5) \\ \\ &-21x&+&15y&=&-24& \\ +&-15x&-&15y&=&60& \\ \midrule &&&\dfrac{-36x}{-36}&=&\dfrac{36}{-36}& \\ \\ &&&x&=&-1& \\ \\ &-3(-1)&-&3y&=&12& \\ &3&-&3y&=&12& \\ &-3&&&&-3& \\ \midrule &&&\dfrac{-3y}{-3}&=&\dfrac{9}{-3}& \\ \\ &&&y&=&-3& \\ (-1,-3)&&&&&& \end{array}$
$\begin{array}{rrrrrrcll} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &&(6x&+&3y&=&-18)&\div &-3 \\ \\ &&-2x&-&y&=&6&& \\ &&+2x&&&&+2x&& \\ \midrule &&&&-y&=&2x&+&6 \\ &&&&y&=&-2x&-&6 \\ \\ 8x&+&7(-2x&-&6)&=&-24&& \\ 8x&-&14x&-&42&=&-24&& \\ &&&+&42&&+42&& \\ \midrule &&&&\dfrac{-6x}{-6}&=&\dfrac{18}{-6}&& \\ \\ &&&&x&=&-3&& \\ \\ &&&&y&=&-2(-3)&-&6 \\ &&&&y&=&6&-&6 \\ &&&&y&=&0&& \\ (-3,0)&&&&&&&& \\ \end{array}$
$\begin{array}{rrrrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &&(-8x&-&8y&=&-8)&\div &(-8) \\ \\ &&x&+&y&=&1&& \\ &&-x&&&&-x&& \\ \midrule &&&&y&=&1&-&x \\ \\ 10x&+&9(1&-&x)&=&1&& \\ 10x&+&9&-&9x&=&1&& \\ &-&9&&&&-9&& \\ \midrule &&&&x&=&-8&& \\ \\ &&&&y&=&1&-&-8 \\ &&&&y&=&9&& \\ (-8,9)&&&&&&&& \end{array}$
$\begin{array}{rrrrrrl} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &(-7x&+&10y&=&13)&(4) \\ &(4x&+&9y&=&22)&(7) \\ \\ &-28x&+&40y&=&52& \\ +&28x&+&63y&=&154& \\ \midrule &&&\dfrac{103y}{103}&=&\dfrac{206}{103}& \\ \\ &&&y&=&2& \\ \\ &4x&+&9(2)&=&22& \\ &4x&+&18&=&22& \\ &&-&18&&-18& \\ \midrule &&&\dfrac{4x}{4}&=&\dfrac{4}{4}& \\ \\ &&&x&=&1& \\ (1,2)&&&&&& \end{array}$

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