Euclid 2020/Problem 1

Problem

1. (a) If $x = 11$ , what is the value of $\frac{3x + 6}{x + 2}$ ?

(b) What is the y-intercept of the line that passes through $A(1, 5)$ and $B(1, 7)$ ?

(c) The lines with equations $y = 3x + 7$ , $y = x + 9$ , and $y = mx + 17$ intersect at a single point. Determine the value of $m$ .

Solution

(a) By plugging in $x=11$ , we can get $\frac{3(11)+6}{11+2}=\frac{39}{13}=3$ . Therefore, our answer is $\boxed{3}$ .

(b) The slope of the line passing through $A(1, 5)$ and $B(1, 7)$ is $\frac{5-7}{1-1}$ which is undefined. So, this is a vertical line, and therefore, there is no y-intercept.

(c) By solving the system of equations $y=3x+7$ and $y=x+9$ , we can get $(x, y)=(1, 10)$ . So, by plugging in $(1, 10)$ into $y=mx+17$ , we get $10=m+17$ , so $m=-7$ . There, our answer is $\boxed{-7}$ .

~Yuhao2012

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Euclid 2020/Problem 1

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