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Algebra Level 2

{ x 3 = a x + 1 4 = b x + 2 43 = a b + 20 4 \large \begin{cases} \dfrac{x}{3}=a \\ \dfrac{x+1}{4}=b \\ \dfrac{x+2}{43}=\dfrac{a-b+20}{4} \end{cases}

If a a , b b , and x x satisfy the system of equations above, find the value of x x .


The answer is 2019.

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2 solutions

Chew-Seong Cheong
Dec 17, 2018

x + 2 43 = a b + 20 4 Given x 3 = a x + 2 43 = x 3 x + 1 4 + 20 4 and x + 1 4 = b = 4 x 3 x 3 + 240 48 = x + 237 48 48 x + 96 = 43 x + 10191 5 x = 10095 x = 2019 \begin{aligned} \frac {x+2}{43} & = \frac {{\color{#3D99F6}a} - {\color{#D61F06}b}+20}4 & \small \color{#3D99F6} \text{Given }\frac x3 = a \\ \frac {x+2}{43} & = \frac {{\color{#3D99F6}\frac x3} - {\color{#D61F06}\frac {x+1}4}+20}4 & \small \color{#D61F06} \text{and }\frac {x+1}4 = b \\ & = \frac {4x-3x-3+240}{48} \\ & = \frac {x+237}{48} \\ 48x + 96 & = 43x + 10191 \\ 5x & = 10095 \\ \implies x & = \boxed{2019} \end{aligned}


Image credit: VectorStock

x = 3 a x=3a

b = 3 a + 1 4 b=\frac{3a+1}{4}

3 a + 2 43 = a 3 a + 1 4 + 20 4 \frac{3a+2}{43}=\frac{a-\frac{3a+1}{4}+20}{4}

So we get a = 673 a=673 , and x = 3 673 = 2019 x=3*673=2019

Happy new year!

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