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Difference between revisions of "Euclid 2020/Problem 1"
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==Solution==
 
==Solution==
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(a) By plugging in <math>x=11</math>, we can get <math>\frac{3(11)+6}{11+2}=\frac{39}{13}=3</math>. Therefore, our answer is <math>\boxed{3}</math>
+
(a) By plugging in <math>x=11</math>, we can get <math>\frac{3(11)+6}{11+2}=\frac{39}{13}=3</math>. Therefore, our answer is <math>\boxed{3}</math>.
 +
 
 
(b) The slope of the line passing through <math>A(1, 5)</math> and <math>B(1, 7)</math> is <math>\frac{5-7}{1-1}</math> which is undefined. So, this is a vertical line, and therefore, there is no y-intercept.
 
(b) The slope of the line passing through <math>A(1, 5)</math> and <math>B(1, 7)</math> is <math>\frac{5-7}{1-1}</math> which is undefined. So, this is a vertical line, and therefore, there is no y-intercept.
 +
 
(c) By solving the system of equations <math>y=3x+7</math> and <math>y=x+9</math>, we can get <math>(x, y)=(1, 10)</math>. So, by plugging in <math>(1, 10)</math> into <math>y=mx+17</math>, we get <math>10=m+17</math>, so <math>m=-7</math>. There, our answer is <math>\boxed{-7}</math>.
 
(c) By solving the system of equations <math>y=3x+7</math> and <math>y=x+9</math>, we can get <math>(x, y)=(1, 10)</math>. So, by plugging in <math>(1, 10)</math> into <math>y=mx+17</math>, we get <math>10=m+17</math>, so <math>m=-7</math>. There, our answer is <math>\boxed{-7}</math>.
 +
 +
~Yuhao2012

Latest revision as of 18:40, 12 October 2025

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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