પૂર્ણાંક સમીકરણો

પૂર્ણાંક સમીકરણો એ ગણિતના બેઝિક મોડેલો છે, જે સંપુર્ણ અથવા પૂર્ણાંક સંખ્યાઓનો ઉપયોગ કરીને સમીકરણોના ઉકેલો શોધવા માટેની પ્રક્રિયાઓને વર્ણવે છે. પૂર્ણાંક સમીકરણોનો ઉપયોગ વિજ્ઞાન, ટેક્નોલોજી, ઇજનેરી અને અન્ય ક્ષેત્રોમાં જોવા મળે છે, ખાસ કરીને જ્યારે સંખ્યાઓની ચોક્કસતા જરૂરી હોય છે. પૂર્ણાંક સમીકરણો વિવિધ પ્રકારના હોઈ શકે છે, જેમ કે લાઈનર, અવિશેષણ પૂર્ણાંક સમીકરણો, અને આ પ્રકારની સમીકરણોને ઉકેલવા માટેના વિવિધ પદ્ધતિઓ છે.

1. પૂર્ણાંક સમીકરણો란 શું છે?

પૂર્ણાંક સમીકરણોમાં એવા સમીકરણોનો સમાવેશ થાય છે જે માત્ર પૂર્ણાંક મૂલ્યો માટે ઉકેલે છે. અર્થાત, સમીકરણનો ઉકેલ સંપૂર્ણ સંખ્યા હશે. આવાં સમીકરણો માટે જે મુખ્ય મુદ્દા છે, તે છે કે દરેક પદ પૂર્ણાંક મૂલ્ય ધરાવતું હોવું જોઈએ.

2. પૂર્ણાંક સમીકરણના ઉકેલ માટે પદ્ધતિઓ:

પૂર્ણાંક સમીકરણોના ઉકેલ માટે કેટલીક પદ્ધતિઓ નીચે મુજબ છે:

2.1. લાઈનર પૂર્ણાંક સમીકરણો

લાઈનર પૂર્ણાંક સમીકરણમાં સમીકરણ સીધી રેખાની સમીકરણ જેવા સ્વરૂપમાં હોય છે. ઉદાહરણ તરીકે:

$ax+b=0$

એમાં $a$ અને $b$ પૂર્ણાંક છે, અને $x$ એ પરિવર્તક છે, જેનો ઉકેલ પૂર્ણાંક હોવો જોઈએ.

Example 1:

$3x+5=11$

Solution:

$3x=11–5$

$3x=6$

$x=\frac{6}{3}$

$=2$

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Example 2:

$2x–4=8$

Solution:

$2x=8+4$

$2x=12$

$x=\frac{12}{2}$

$=6$

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Example 3:

$5x+7=22$

Solution:

$5x=22–7$

$5x=15$

$x=\frac{15}{5}$

$=3$

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Example 4:

$4x+10=18$

Solution:

$4x=18–10$

$4x=8$

$x=\frac{8}{4}$

$=2$

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Example 5:

$7x–3=18$

Solution:

$7x=18+3$

$7x=21$

$x=\frac{21}{7}$

$=3$

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2.2. અવિશેષણ પૂર્ણાંક સમીકરણો

અવિશેષણ પૂર્ણાંક સમીકરણ એ તે સમીકરણ છે, જેમાં $x$ અને $y$ જેવી વિવિધ પેરામીટર્સના મિશ્રણમાં ઉકેલો પૂર્ણાંક હોવો જોઈએ. ઉદાહરણ તરીકે:

$ax+by=c$

જેમાં $a,b$ અને $c$ પૂર્ણાંક છે, અને ઉકેલ પણ પૂર્ણાંક સંખ્યાઓમાં જ હોય.

Example 6:

$2x+3y=12$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$2(0)+3y=12$

$3y=12$

$y=\frac{12}{3}$

$=4$

For $x=3$:

$2(3)+3y=12$

$6+3y=12$

$3y=12–6$

$y=\frac{6}{3}$

$=2$

Thus, $(x,y)=(0,4),(3,2)$ are solutions.

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Example 7:

$3x+4y=24$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$3(0)+4y=24$

$4y=24$

$y=\frac{24}{4}$

$=6$

For $x=4$:

$3(4)+4y=24$

$12+4y=24$

$4y=24–12$

$y=\frac{12}{4}$

$=3$

Thus, $(x,y)=(0,6),(4,3)$ are solutions.

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Example 8:

$5x+2y=10$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$5(0)+2y=10$

$2y=10$

$y=\frac{10}{2}$

$=5$

For $x=2$:

$5(2)+2y=10$

$10+2y=10$

$2y=10–10$

$y=0$

Thus, $(x,y)=(0,5),(2,0)$ are solutions.

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Example 9:

$6x–3y=9$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$6(0)–3y=9$

$-3y=9$

$y=\frac{9}{-3}$

$=-3$

For $x=3$:

$6(3)–3y=9$

$18–3y=9$

$-3y=9–18$

$y=\frac{-9}{-3}$

$=3$

Thus, $(x,y)=(0,-3),(3,3)$ are solutions.

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Example 10:

$4x+7y=28$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$4(0)+7y=28$

$7y=28$

$y=\frac{28}{7}$

$=4$

For $x=7$:

$4(7)+7y=28$

$28+7y=28$

$7y=28–28$

$y=0$

Thus, $(x,y)=(0,4),(7,0)$ are solutions.

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Example 11:

$3x+5y=15$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$3(0)+5y=15$

$5y=15$

$y=\frac{15}{5}$

$=3$

For $x=5$:

$3(5)+5y=15$

$15+5y=15$

$5y=15–15$

$y=0$

Thus, $(x,y)=(0,3),(5,0)$ are solutions.

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Example 12:

$7x+2y=14$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$7(0)+2y=14$

$2y=14$

$y=\frac{14}{2}$

$=7$

For $x=2$:

$7(2)+2y=14$

$14+2y=14$

$2y=14–14$

$y=0$

Thus, $(x,y)=(0,7),(2,0)$ are solutions.

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Example 13:

$6x+8y=24$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$6(0)+8y=24$

$8y=24$

$y=\frac{24}{8}$

$=3$

For $x=4$:

$6(4)+8y=24$

$24+8y=24$

$8y=24–24$

$y=0$

Thus, $(x,y)=(0,3),(4,0)$ are solutions.

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Example 14:

$5x+9y=45$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$5(0)+9y=45$

$9y=45$

$y=\frac{45}{9}$

$=5$

For $x=9$:

$5(9)+9y=45$

$45+9y=45$

$9y=45–45$

$y=0$

Thus, $(x,y)=(0,5),(9,0)$ are solutions.

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Example 15:

$4x+6y=12$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$4(0)+6y=12$

$6y=12$

$y=\frac{12}{6}$

$=2$

For $x=3$:

$4(3)+6y=12$

$12+6y=12$

$6y=12–12$

$y=0$

Thus, $(x,y)=(0,2),(3,0)$ are solutions.

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Example 16:

$2x+4y=20$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$2(0)+4y=20$

$4y=20$

$y=\frac{20}{4}$

$=5$

For $x=10$:

$2(10)+4y=20$

$20+4y=20$

$4y=20–20$

$y=0$

Thus, $(x,y)=(0,5),(10,0)$ are solutions.

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Example 17:

$3x+6y=18$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$3(0)+6y=18$

$6y=18$

$y=\frac{18}{6}$

$=3$

For $x=6$:

$3(6)+6y=18$

$18+6y=18$

$6y=18–18$

$y=0$

Thus, $(x,y)=(0,3),(6,0)$ are solutions.

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Example 18:

$8x+10y=40$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$8(0)+10y=40$

$10y=40$

$y=\frac{40}{10}$

$=4$

For $x=5$:

$8(5)+10y=40$

$40+10y=40$

$10y=40–40$

$y=0$

Thus, $(x,y)=(0,4),(5,0)$ are solutions.

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Example 19:

$9x+12y=36$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$9(0)+12y=36$

$12y=36$

$y=\frac{36}{12}$

$=3$

For $x=4$:

$9(4)+12y=36$

$36+12y=36$

$12y=36–36$

$y=0$

Thus, $(x,y)=(0,3),(4,0)$ are solutions.

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Example 20:

$5x+8y=40$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$5(0)+8y=40$

$8y=40$

$y=\frac{40}{8}$

$=5$

For $x=8$:

$5(8)+8y=40$

$40+8y=40$

$8y=40–40$

$y=0$

Thus, $(x,y)=(0,5),(8,0)$ are solutions.

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Example 21:

$6x+9y=54$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$6(0)+9y=54$

$9y=54$

$y=\frac{54}{9}$

$=6$

For $x=9$:

$6(9)+9y=54$

$54+9y=54$

$9y=54–54$

$y=0$

Thus, $(x,y)=(0,6),(9,0)$ are solutions.

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Example 22:

$7x+5y=35$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$7(0)+5y=35$

$5y=35$

$y=\frac{35}{5}$

$=7$

For $x=5$:

$7(5)+5y=35$

$35+5y=35$

$5y=35–35$

$y=0$

Thus, $(x,y)=(0,7),(5,0)$ are solutions.

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Example 23:

$4x+7y=28$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$4(0)+7y=28$

$7y=28$

$y=\frac{28}{7}$

$=4$

For $x=7$:

$4(7)+7y=28$

$28+7y=28$

$7y=28–28$

$y=0$

Thus, $(x,y)=(0,4),(7,0)$ are solutions.

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Example 24:

$8x+6y=48$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$8(0)+6y=48$

$6y=48$

$y=\frac{48}{6}$

$=8$

For $x=6$:

$8(6)+6y=48$

$48+6y=48$

$6y=48–48$

$y=0$

Thus, $(x,y)=(0,8),(6,0)$ are solutions.

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Example 25:

$10x+15y=60$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$10(0)+15y=60$

$15y=60$

$y=\frac{60}{15}$

$=4$

For $x=6$:

$10(6)+15y=60$

$60+15y=60$

$15y=60–60$

$y=0$

Thus, $(x,y)=(0,4),(6,0)$ are solutions.

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Example 26:

$9x+3y=27$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$9(0)+3y=27$

$3y=27$

$y=\frac{27}{3}$

$=9$

For $x=3$:

$9(3)+3y=27$

$27+3y=27$

$3y=27–27$

$y=0$

Thus, $(x,y)=(0,9),(3,0)$ are solutions.

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Example 27:

$6x+4y=24$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$6(0)+4y=24$

$4y=24$

$y=\frac{24}{4}$

$=6$

For $x=4$:

$6(4)+4y=24$

$24+4y=24$

$4y=24–24$

$y=0$

Thus, $(x,y)=(0,6),(4,0)$ are solutions.

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Example 28:

$5x+12y=60$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$5(0)+12y=60$

$12y=60$

$y=\frac{60}{12}$

$=5$

For $x=12$:

$5(12)+12y=60$

$60+12y=60$

$12y=60–60$

$y=0$

Thus, $(x,y)=(0,5),(12,0)$ are solutions.

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Example 29:

$11x+9y=99$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$11(0)+9y=99$

$9y=99$

$y=\frac{99}{9}$

$=11$

For $x=9$:

$11(9)+9y=99$

$99+9y=99$

$9y=99–99$

$y=0$

Thus, $(x,y)=(0,11),(9,0)$ are solutions.

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Example 30:

$8x+14y=56$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$8(0)+14y=56$

$14y=56$

$y=\frac{56}{14}$

$=4$

For $x=7$:

$8(7)+14y=56$

$56+14y=56$

$14y=56–56$

$y=0$

Thus, $(x,y)=(0,4),(7,0)$ are solutions.

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Example 31:

$7x+5y=35$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$7(0)+5y=35$

$5y=35$

$y=\frac{35}{5}$

$=7$

For $x=5$:

$7(5)+5y=35$

$35+5y=35$

$5y=35–35$

$y=0$

Thus, $(x,y)=(0,7),(5,0)$ are solutions.

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Example 32:

$4x+9y=36$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$4(0)+9y=36$

$9y=36$

$y=\frac{36}{9}$

$=4$

For $x=9$:

$4(9)+9y=36$

$36+9y=36$

$9y=36–36$

$y=0$

Thus, $(x,y)=(0,4),(9,0)$ are solutions.

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Example 33:

$5x+10y=50$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$5(0)+10y=50$

$10y=50$

$y=\frac{50}{10}$

$=5$

For $x=10$:

$5(10)+10y=50$

$50+10y=50$

$10y=50–50$

$y=0$

Thus, $(x,y)=(0,5),(10,0)$ are solutions.

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Example 34:

$6x+3y=18$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$6(0)+3y=18$

$3y=18$

$y=\frac{18}{3}$

$=6$

For $x=3$:

$6(3)+3y=18$

$18+3y=18$

$3y=18–18$

$y=0$

Thus, $(x,y)=(0,6),(3,0)$ are solutions.

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Example 35:

$8x+5y=40$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$8(0)+5y=40$

$5y=40$

$y=\frac{40}{5}$

$=8$

For $x=5$:

$8(5)+5y=40$

$40+5y=40$

$5y=40–40$

$y=0$

Thus, $(x,y)=(0,8),(5,0)$ are solutions.

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Example 36:

$9x+12y=72$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$9(0)+12y=72$

$12y=72$

$y=\frac{72}{12}$

$=6$

For $x=8$:

$9(8)+12y=72$

$72+12y=72$

$12y=72–72$

$y=0$

Thus, $(x,y)=(0,6),(8,0)$ are solutions.

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Example 37:

$10x+7y=70$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$10(0)+7y=70$

$7y=70$

$y=\frac{70}{7}$

$=10$

For $x=7$:

$10(7)+7y=70$

$70+7y=70$

$7y=70–70$

$y=0$

Thus, $(x,y)=(0,10),(7,0)$ are solutions.

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Example 38:

$11x+9y=99$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$11(0)+9y=99$

$9y=99$

$y=\frac{99}{9}$

$=11$

For $x=9$:

$11(9)+9y=99$

$99+9y=99$

$9y=99–99$

$y=0$

Thus, $(x,y)=(0,11),(9,0)$ are solutions.

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Example 39:

$5x+6y=30$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$5(0)+6y=30$

$6y=30$

$y=\frac{30}{6}$

$=5$

For $x=6$:

$5(6)+6y=30$

$30+6y=30$

$6y=30–30$

$y=0$

Thus, $(x,y)=(0,5),(6,0)$ are solutions.

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Example 40:

$12x+8y=96$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$12(0)+8y=96$

$8y=96$

$y=\frac{96}{8}$

$=12$

For $x=8$:

$12(8)+8y=96$

$96+8y=96$

$8y=96–96$

$y=0$

Thus, $(x,y)=(0,12),(8,0)$ are solutions.

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Example 41:

$8x+4y=32$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$8(0)+4y=32$

$4y=32$

$y=\frac{32}{4}$

$=8$

For $x=4$:

$8(4)+4y=32$

$32+4y=32$

$4y=32–32$

$y=0$

Thus, $(x,y)=(0,8),(4,0)$ are solutions.

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Example 42:

$7x+3y=21$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$7(0)+3y=21$

$3y=21$

$y=\frac{21}{3}$

$=7$

For $x=3$:

$7(3)+3y=21$

$21+3y=21$

$3y=21–21$

$y=0$

Thus, $(x,y)=(0,7),(3,0)$ are solutions.

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Example 43:

$9x+4y=36$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$9(0)+4y=36$

$4y=36$

$y=\frac{36}{4}$

$=9$

For $x=4$:

$9(4)+4y=36$

$36+4y=36$

$4y=36–36$

$y=0$

Thus, $(x,y)=(0,9),(4,0)$ are solutions.

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Example 44:

$5x+7y=35$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$5(0)+7y=35$

$7y=35$

$y=\frac{35}{7}$

$=5$

For $x=7$:

$5(7)+7y=35$

$35+7y=35$

$7y=35–35$

$y=0$

Thus, $(x,y)=(0,5),(7,0)$ are solutions.

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Example 45:

$6x+11y=66$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$6(0)+11y=66$

$11y=66$

$y=\frac{66}{11}$

$=6$

For $x=11$:

$6(11)+11y=66$

$66+11y=66$

$11y=66–66$

$y=0$

Thus, $(x,y)=(0,6),(11,0)$ are solutions.

---

Example 46:

$3x+6y=30$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$3(0)+6y=30$

$6y=30$

$y=\frac{30}{6}$

$=5$

For $x=6$:

$3(6)+6y=30$

$18+6y=30$

$6y=30–18$

$y=\frac{12}{6}$

$=2$

Thus, $(x,y)=(0,5),(6,2)$ are solutions.

---

Example 47:

$10x+8y=40$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$10(0)+8y=40$

$8y=40$

$y=\frac{40}{8}$

$=5$

For $x=4$:

$10(4)+8y=40$

$40+8y=40$

$8y=40–40$

$y=0$

Thus, $(x,y)=(0,5),(4,0)$ are solutions.

---

Example 48:

$7x+8y=56$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$7(0)+8y=56$

$8y=56$

$y=\frac{56}{8}$

$=7$

For $x=8$:

$7(8)+8y=56$

$56+8y=56$

$8y=56–56$

$y=0$

Thus, $(x,y)=(0,7),(8,0)$ are solutions.

---

Example 49:

$12x+5y=60$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$12(0)+5y=60$

$5y=60$

$y=\frac{60}{5}$

$=12$

For $x=5$:

$12(5)+5y=60$

$60+5y=60$

$5y=60–60$

$y=0$

Thus, $(x,y)=(0,12),(5,0)$ are solutions.

---

Example 50:

$9x+11y=99$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$9(0)+11y=99$

$11y=99$

$y=\frac{99}{11}$

$=9$

For $x=11$:

$9(11)+11y=99$

$99+11y=99$

$11y=99–99$

$y=0$

Thus, $(x,y)=(0,9),(11,0)$ are solutions.

---

Example 51:

$5x+4y=20$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$5(0)+4y=20$

$4y=20$

$y=\frac{20}{4}$

$=5$

For $x=4$:

$5(4)+4y=20$

$20+4y=20$

$4y=20–20$

$y=0$

Thus, $(x,y)=(0,5),(4,0)$ are solutions.

---

Example 52:

$8x+10y=80$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$8(0)+10y=80$

$10y=80$

$y=\frac{80}{10}$

$=8$

For $x=10$:

$8(10)+10y=80$

$80+10y=80$

$10y=80–80$

$y=0$

Thus, $(x,y)=(0,8),(10,0)$ are solutions.

---

Example 53:

$3x+2y=12$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$3(0)+2y=12$

$2y=12$

$y=\frac{12}{2}$

$=6$

For $x=4$:

$3(4)+2y=12$

$12+2y=12$

$2y=12–12$

$y=0$

Thus, $(x,y)=(0,6),(4,0)$ are solutions.

---

Example 54:

$6x+5y=30$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$6(0)+5y=30$

$5y=30$

$y=\frac{30}{5}$

$=6$

For $x=5$:

$6(5)+5y=30$

$30+5y=30$

$5y=30–30$

$y=0$

Thus, $(x,y)=(0,6),(5,0)$ are solutions.

---

Example 55:

$7x+3y=21$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$7(0)+3y=21$

$3y=21$

$y=\frac{21}{3}$

$=7$

For $x=3$:

$7(3)+3y=21$

$21+3y=21$

$3y=21–21$

$y=0$

Thus, $(x,y)=(0,7),(3,0)$ are solutions.

---

Example 56:

$9x+6y=54$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$9(0)+6y=54$

$6y=54$

$y=\frac{54}{6}$

$=9$

For $x=6$:

$9(6)+6y=54$

$54+6y=54$

$6y=54–54$

$y=0$

Thus, $(x,y)=(0,9),(6,0)$ are solutions.

---

Example 57:

$4x+10y=40$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$4(0)+10y=40$

$10y=40$

$y=\frac{40}{10}$

$=4$

For $x=10$:

$4(10)+10y=40$

$40+10y=40$

$10y=40–40$

$y=0$

Thus, $(x,y)=(0,4),(10,0)$ are solutions.

---

Example 58:

$5x+9y=45$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$5(0)+9y=45$

$9y=45$

$y=\frac{45}{9}$

$=5$

For $x=9$:

$5(9)+9y=45$

$45+9y=45$

$9y=45–45$

$y=0$

Thus, $(x,y)=(0,5),(9,0)$ are solutions.

---

Example 59:

$6x+8y=48$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$6(0)+8y=48$

$8y=48$

$y=\frac{48}{8}$

$=6$

For $x=8$:

$6(8)+8y=48$

$48+8y=48$

$8y=48–48$

$y=0$

Thus, $(x,y)=(0,6),(8,0)$ are solutions.

---

Example 60:

$10x+7y=70$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$10(0)+7y=70$

$7y=70$

$y=\frac{70}{7}$

$=10$

For $x=7$:

$10(7)+7y=70$

$70+7y=70$

$7y=70–70$

$y=0$

Thus, $(x,y)=(0,10),(7,0)$ are solutions.

---

Example 61:

$4x+6y=24$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$4(0)+6y=24$

$6y=24$

$y=\frac{24}{6}$

$=4$

For $x=6$:

$4(6)+6y=24$

$24+6y=24$

$6y=24–24$

$y=0$

Thus, $(x,y)=(0,4),(6,0)$ are solutions.

---

Example 62:

$5x+12y=60$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$5(0)+12y=60$

$12y=60$

$y=\frac{60}{12}$

$=5$

For $x=12$:

$5(12)+12y=60$

$60+12y=60$

$12y=60–60$

$y=0$

Thus, $(x,y)=(0,5),(12,0)$ are solutions.

---

Example 63:

$8x+9y=72$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$8(0)+9y=72$

$9y=72$

$y=\frac{72}{9}$

$=8$

For $x=9$:

$8(9)+9y=72$

$72+9y=72$

$9y=72–72$

$y=0$

Thus, $(x,y)=(0,8),(9,0)$ are solutions.

---

Example 64:

$11x+5y=55$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$11(0)+5y=55$

$5y=55$

$y=\frac{55}{5}$

$=11$

For $x=5$:

$11(5)+5y=55$

$55+5y=55$

$5y=55–55$

$y=0$

Thus, $(x,y)=(0,11),(5,0)$ are solutions.

---

Example 65:

$7x+8y=56$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$7(0)+8y=56$

$8y=56$

$y=\frac{56}{8}$

$=7$

For $x=8$:

$7(8)+8y=56$

$56+8y=56$

$8y=56–56$

$y=0$

Thus, $(x,y)=(0,7),(8,0)$ are solutions.

---

Example 66:

$6x+14y=84$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$6(0)+14y=84$

$14y=84$

$y=\frac{84}{14}$

$=6$

For $x=14$:

$6(14)+14y=84$

$84+14y=84$

$14y=84–84$

$y=0$

Thus, $(x,y)=(0,6),(14,0)$ are solutions.

---

Example 67:

$12x+9y=108$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$12(0)+9y=108$

$9y=108$

$y=\frac{108}{9}$

$=12$

For $x=9$:

$12(9)+9y=108$

$108+9y=108$

$9y=108–108$

$y=0$

Thus, $(x,y)=(0,12),(9,0)$ are solutions.

---

Example 68:

$10x+6y=60$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$10(0)+6y=60$

$6y=60$

$y=\frac{60}{6}$

$=10$

For $x=6$:

$10(6)+6y=60$

$60+6y=60$

$6y=60–60$

$y=0$

Thus, $(x,y)=(0,10),(6,0)$ are solutions.

---

Example 69:

$9x+5y=45$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$9(0)+5y=45$

$5y=45$

$y=\frac{45}{5}$

$=9$

For $x=5$:

$9(5)+5y=45$

$45+5y=45$

$5y=45–45$

$y=0$

Thus, $(x,y)=(0,9),(5,0)$ are solutions.

---

Example 70:

$4x+11y=44$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$4(0)+11y=44$

$11y=44$

$y=\frac{44}{11}$

$=4$

For $x=11$:

$4(11)+11y=44$

$44+11y=44$

$11y=44–44$

$y=0$

Thus, $(x,y)=(0,4),(11,0)$ are solutions.

---

Example 71:

$6x+8y=48$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$6(0)+8y=48$

$8y=48$

$y=\frac{48}{8}$

$=6$

For $x=8$:

$6(8)+8y=48$

$48+8y=48$

$8y=48–48$

$y=0$

Thus, $(x,y)=(0,6),(8,0)$ are solutions.

---

Example 72:

$7x+11y=77$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$7(0)+11y=77$

$11y=77$

$y=\frac{77}{11}$

$=7$

For $x=11$:

$7(11)+11y=77$

$77+11y=77$

$11y=77–77$

$y=0$

Thus, $(x,y)=(0,7),(11,0)$ are solutions.

---

Example 73:

$8x+6y=48$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$8(0)+6y=48$

$6y=48$

$y=\frac{48}{6}$

$=8$

For $x=6$:

$8(6)+6y=48$

$48+6y=48$

$6y=48–48$

$y=0$

Thus, $(x,y)=(0,8),(6,0)$ are solutions.

---

Example 74:

$5x+9y=45$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$5(0)+9y=45$

$9y=45$

$y=\frac{45}{9}$

$=5$

For $x=9$:

$5(9)+9y=45$

$45+9y=45$

$9y=45–45$

$y=0$

Thus, $(x,y)=(0,5),(9,0)$ are solutions.

---

Example 75:

$11x+4y=44$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$11(0)+4y=44$

$4y=44$

$y=\frac{44}{4}$

$=11$

For $x=4$:

$11(4)+4y=44$

$44+4y=44$

$4y=44–44$

$y=0$

Thus, $(x,y)=(0,11),(4,0)$ are solutions.

---

Example 76:

$12x+9y=108$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$12(0)+9y=108$

$9y=108$

$y=\frac{108}{9}$

$=12$

For $x=9$:

$12(9)+9y=108$

$108+9y=108$

$9y=108–108$

$y=0$

Thus, $(x,y)=(0,12),(9,0)$ are solutions.

---

Example 77:

$4x+7y=28$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$4(0)+7y=28$

$7y=28$

$y=\frac{28}{7}$

$=4$

For $x=7$:

$4(7)+7y=28$

$28+7y=28$

$7y=28–28$

$y=0$

Thus, $(x,y)=(0,4),(7,0)$ are solutions.

---

Example 78:

$9x+10y=90$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$9(0)+10y=90$

$10y=90$

$y=\frac{90}{10}$

$=9$

For $x=10$:

$9(10)+10y=90$

$90+10y=90$

$10y=90–90$

$y=0$

Thus, $(x,y)=(0,9),(10,0)$ are solutions.

---

Example 79:

$8x+5y=40$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$8(0)+5y=40$

$5y=40$

$y=\frac{40}{5}$

$=8$

For $x=5$:

$8(5)+5y=40$

$40+5y=40$

$5y=40–40$

$y=0$

Thus, $(x,y)=(0,8),(5,0)$ are solutions.

---

Example 80:

$6x+10y=60$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$6(0)+10y=60$

$10y=60$

$y=\frac{60}{10}$

$=6$

For $x=10$:

$6(10)+10y=60$

$60+10y=60$

$10y=60–60$

$y=0$

Thus, $(x,y)=(0,6),(10,0)$ are solutions.

---

Example 81:

$7x+6y=42$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$7(0)+6y=42$

$6y=42$

$y=\frac{42}{6}$

$=7$

For $x=6$:

$7(6)+6y=42$

$42+6y=42$

$6y=42–42$

$y=0$

Thus, $(x,y)=(0,7),(6,0)$ are solutions.

---

Example 82:

$5x+10y=50$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$5(0)+10y=50$

$10y=50$

$y=\frac{50}{10}$

$=5$

For $x=10$:

$5(10)+10y=50$

$50+10y=50$

$10y=50–50$

$y=0$

Thus, $(x,y)=(0,5),(10,0)$ are solutions.

---

Example 83:

$6x+11y=66$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$6(0)+11y=66$

$11y=66$

$y=\frac{66}{11}$

$=6$

For $x=11$:

$6(11)+11y=66$

$66+11y=66$

$11y=66–66$

$y=0$

Thus, $(x,y)=(0,6),(11,0)$ are solutions.

---

Example 84:

$9x+4y=36$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$9(0)+4y=36$

$4y=36$

$y=\frac{36}{4}$

$=9$

For $x=4$:

$9(4)+4y=36$

$36+4y=36$

$4y=36–36$

$y=0$

Thus, $(x,y)=(0,9),(4,0)$ are solutions.

---

Example 85:

$8x+12y=72$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$8(0)+12y=72$

$12y=72$

$y=\frac{72}{12}$

$=6$

For $x=12$:

$8(12)+12y=72$

$72+12y=72$

$12y=72–72$

$y=0$

Thus, $(x,y)=(0,6),(12,0)$ are solutions.

---

Example 86:

$10x+6y=60$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$10(0)+6y=60$

$6y=60$

$y=\frac{60}{6}$

$=10$

For $x=6$:

$10(6)+6y=60$

$60+6y=60$

$6y=60–60$

$y=0$

Thus, $(x,y)=(0,10),(6,0)$ are solutions.

---

Example 87:

$7x+8y=56$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$7(0)+8y=56$

$8y=56$

$y=\frac{56}{8}$

$=7$

For $x=8$:

$7(8)+8y=56$

$56+8y=56$

$8y=56–56$

$y=0$

Thus, $(x,y)=(0,7),(8,0)$ are solutions.

---

Example 88:

$6x+5y=30$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$6(0)+5y=30$

$5y=30$

$y=\frac{30}{5}$

$=6$

For $x=5$:

$6(5)+5y=30$

$30+5y=30$

$5y=30–30$

$y=0$

Thus, $(x,y)=(0,6),(5,0)$ are solutions.

---

Example 89:

$5x+6y=30$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$5(0)+6y=30$

$6y=30$

$y=\frac{30}{6}$

$=5$

For $x=6$:

$5(6)+6y=30$

$30+6y=30$

$6y=30–30$

$y=0$

Thus, $(x,y)=(0,5),(6,0)$ are solutions.

---

Example 90:

$8x+9y=72$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$8(0)+9y=72$

$9y=72$

$y=\frac{72}{9}$

$=8$

For $x=9$:

$8(9)+9y=72$

$72+9y=72$

$9y=72–72$

$y=0$

Thus, $(x,y)=(0,8),(9,0)$ are solutions.

---

Example 91:

$4x+8y=32$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$4(0)+8y=32$

$8y=32$

$y=\frac{32}{8}$

$=4$

For $x=8$:

$4(8)+8y=32$

$32+8y=32$

$8y=32–32$

$y=0$

Thus, $(x,y)=(0,4),(8,0)$ are solutions.

---

Example 92:

$6x+4y=24$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$6(0)+4y=24$

$4y=24$

$y=\frac{24}{4}$

$=6$

For $x=4$:

$6(4)+4y=24$

$24+4y=24$

$4y=24–24$

$y=0$

Thus, $(x,y)=(0,6),(4,0)$ are solutions.

---

Example 93:

$10x+7y=70$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$10(0)+7y=70$

$7y=70$

$y=\frac{70}{7}$

$=10$

For $x=7$:

$10(7)+7y=70$

$70+7y=70$

$7y=70–70$

$y=0$

Thus, $(x,y)=(0,10),(7,0)$ are solutions.

---

Example 94:

$8x+5y=40$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$8(0)+5y=40$

$5y=40$

$y=\frac{40}{5}$

$=8$

For $x=5$:

$8(5)+5y=40$

$40+5y=40$

$5y=40–40$

$y=0$

Thus, $(x,y)=(0,8),(5,0)$ are solutions.

---

Example 95:

$9x+8y=72$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$9(0)+8y=72$

$8y=72$

$y=\frac{72}{8}$

$=9$

For $x=8$:

$9(8)+8y=72$

$72+8y=72$

$8y=72–72$

$y=0$

Thus, $(x,y)=(0,9),(8,0)$ are solutions.

---

Example 96:

$12x+6y=72$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$12(0)+6y=72$

$6y=72$

$y=\frac{72}{6}$

$=12$

For $x=6$:

$12(6)+6y=72$

$72+6y=72$

$6y=72–72$

$y=0$

Thus, $(x,y)=(0,12),(6,0)$ are solutions.

---

Example 97:

$7x+9y=63$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$7(0)+9y=63$

$9y=63$

$y=\frac{63}{9}$

$=7$

For $x=9$:

$7(9)+9y=63$

$63+9y=63$

$9y=63–63$

$y=0$

Thus, $(x,y)=(0,7),(9,0)$ are solutions.

---

Example 98:

$4x+6y=24$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$4(0)+6y=24$

$6y=24$

$y=\frac{24}{6}$

$=4$

For $x=6$:

$4(6)+6y=24$

$24+6y=24$

$6y=24–24$

$y=0$

Thus, $(x,y)=(0,4),(6,0)$ are solutions.

---

Example 99:

$5x+8y=40$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$5(0)+8y=40$

$8y=40$

$y=\frac{40}{8}$

$=5$

For $x=8$:

$5(8)+8y=40$

$40+8y=40$

$8y=40–40$

$y=0$

Thus, $(x,y)=(0,5),(8,0)$ are solutions.

---

Example 100:

$10x+4y=40$

Solution:

For different values of $x$, let’s find $y$:

For $x=0$:

$10(0)+4y=40$

$4y=40$

$y=\frac{40}{4}$

$=10$

For $x=4$:

$10(4)+4y=40$

$40+4y=40$

$4y=40–40$

$y=0$

Thus, $(x,y)=(0,10),(4,0)$ are solutions.