Line $GH$ passes through points $(2,5)$ and $(6,9)$. Which equation represents line $GH$?
A. $y=x+3$
B. $y=x-3$
C. $y=3x+3$
D. $y=3x-3$
Answer (1)
Calculate the slope using the formula: m = x 2 − x 1 y 2 − y 1 = 6 − 2 9 − 5 = 1 .
Use the point-slope form with point ( 2 , 5 ) : y − 5 = 1 ( x − 2 ) .
Convert to slope-intercept form: y = x − 2 + 5 .
The equation of the line is y = x + 3 .
Explanation
Problem Analysis We are given two points, ( 2 , 5 ) and ( 6 , 9 ) , through which line G H passes. Our goal is to find the equation of this line. We will first calculate the slope of the line and then use the point-slope form to determine the equation. Finally, we will convert it to slope-intercept form to match the given options.
Calculating the Slope The slope m of a line passing through two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by the formula: m = x 2 − x 1 y 2 − y 1 In our case, ( x 1 , y 1 ) = ( 2 , 5 ) and ( x 2 , y 2 ) = ( 6 , 9 ) . Plugging in these values, we get: m = 6 − 2 9 − 5 = 4 4 = 1 So, the slope of line G H is 1.
Using Point-Slope Form Now that we have the slope, we can use the point-slope form of a line, which is given by: y − y 1 = m ( x − x 1 ) Using the point ( 2 , 5 ) and the slope m = 1 , we have: y − 5 = 1 ( x − 2 ) y − 5 = x − 2
Converting to Slope-Intercept Form To convert the equation to slope-intercept form ( y = m x + b ), we solve for y :
y = x − 2 + 5 y = x + 3 Thus, the equation of line G H is y = x + 3 .
Final Answer Comparing our result with the given options, we find that the equation y = x + 3 matches the first option.
Examples
Understanding linear equations is crucial in many real-world applications. For instance, consider a scenario where a taxi charges a fixed fee plus a per-mile rate. If the taxi charges a fixed fee of $3 and a per-mile rate of 1 , t h e t o t a l cos t y f or a r i d eo f x mi l esc anb ere p rese n t e d b y t h ee q u a t i o n y = x + 3$. This is a linear equation, and by understanding how to derive and interpret such equations, you can easily calculate the cost of any taxi ride based on the distance traveled. Similarly, linear equations are used in various fields like physics, economics, and computer science to model relationships between variables.
