Difference between revisions of "Euclid 2020/Problem 1"

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 $\boxed{-7}$ .

@@ Line 1: / Line 1: @@
+==Problem==
 .
 (a) If <math>x = 11</math>, what is the value of <math>\frac{3x + 6}{x + 2}</math>?
@@ Line 6: / Line 7: @@
 (c) The lines with equations <math>y = 3x + 7</math>, <math>y = x + 9</math>, and <math>y = mx + 17</math> intersect at a
 single point. Determine the value of <math>m</math>.
+==Solution==
+(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.
+(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>.

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Difference between revisions of "Euclid 2020/Problem 1"

Revision as of 18:40, 12 October 2025

Problem

Solution