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 (2)
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 , which simplifies to y = x + 3 .
The equation representing line GH is y = x + 3 .
Explanation
Understanding the Problem We are given two points, (2, 5) and (6, 9), and we need to find the equation of the line that passes through them.
Calculating the Slope First, we need to find the slope of the line. The slope, denoted by m , is calculated as the change in y divided by the change in x :
m = x 2 − x 1 y 2 − y 1 Using the given points (2, 5) and (6, 9), we have: m = 6 − 2 9 − 5 = 4 4 = 1 So, the slope of the line is 1.
Using the Point-Slope Form Now that we have the slope, we can use the point-slope form of a linear equation, which is: y − y 1 = m ( x − x 1 ) We can use either of the given points. Let's use (2, 5). Substituting the slope m = 1 and the point (2, 5) into the equation, we get: y − 5 = 1 ( x − 2 ) y − 5 = x − 2
Converting to Slope-Intercept Form Next, we need to convert the equation to the slope-intercept form, which is y = m x + b , where b is the y-intercept. To do this, we solve for y :
y = x − 2 + 5 y = x + 3 So, the equation of the line is y = x + 3 .
Final Answer Comparing our equation y = x + 3 with the given options, we see that it matches option O y=x+3. Therefore, the equation that represents line GH is y = x + 3 .
Examples
Understanding linear equations is crucial in many real-world scenarios. For instance, consider a taxi service that charges a fixed fee plus a per-mile rate. If the initial fee is $3 and the per-mile rate is 1 , t h e t o t a l cos t y f or a r i d eo f x mi l esc anb e m o d e l e d b y t h ee q u a t i o n y = x + 3$. This is the same form as the equation we found for line GH. Similarly, in physics, the distance traveled at a constant speed can be modeled using a linear equation, where the slope represents the speed and the y-intercept represents the initial position.
The equation of line GH that passes through the points (2, 5) and (6, 9) is y = x + 3. This corresponds to option A. We found this by calculating the slope and using the point-slope form to derive the equation.
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