Line $QR$ goes through points $Q(0,1)$ and $R(2,7)$. Which equation represents line $QR$?
A. $y-1=6x$
B. $y-1=3x$
C. $y-7=2x-6$
D. $y-7=x-2$
Answer (2)
Calculate the slope of the line using the formula m = x 2 − x 1 y 2 − y 1 , where ( x 1 , y 1 ) = ( 0 , 1 ) and ( x 2 , y 2 ) = ( 2 , 7 ) , which gives m = 3 .
Use the point-slope form of a linear equation: y − y 1 = m ( x − x 1 ) .
Substitute the coordinates of point Q ( 0 , 1 ) and the slope m = 3 into the point-slope form: y − 1 = 3 ( x − 0 ) .
Simplify the equation to get the final answer: y − 1 = 3 x .
Explanation
Understanding the Problem The problem asks us to find the equation of a line that passes through two given points, Q ( 0 , 1 ) and R ( 2 , 7 ) . We will use the point-slope form of a linear equation to find the equation of the line.
Calculating the Slope First, we need to calculate the slope of the line. The slope m is given by the formula: m = x 2 − x 1 y 2 − y 1 where ( x 1 , y 1 ) and ( x 2 , y 2 ) are the coordinates of the two points. In our case, ( x 1 , y 1 ) = ( 0 , 1 ) and ( x 2 , y 2 ) = ( 2 , 7 ) . Plugging these values into the formula, we get: m = 2 − 0 7 − 1 = 2 6 = 3
Applying 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 point Q or point R for ( x 1 , y 1 ) . Let's use point Q ( 0 , 1 ) . Substituting the values m = 3 , x 1 = 0 , and y 1 = 1 into the point-slope form, we get: y − 1 = 3 ( x − 0 ) y − 1 = 3 x
Finding the Matching Equation Now, let's check if our equation matches any of the given options. The equation y − 1 = 3 x is one of the options.
Final Answer Therefore, the equation that represents line QR is y − 1 = 3 x .
Examples
Understanding linear equations is crucial in many real-world applications. For instance, consider a taxi service that charges a fixed fee plus a per-mile rate. If the fixed fee is $1 and the per-mile rate is 3 , t h e t o t a l cos t y f or a r i d eo f x mi l esc anb ere p rese n t e d b y t h ee q u a t i o n y = 3x + 1$. This is a linear equation similar to the one we found. By understanding linear equations, we can easily calculate the cost of a taxi ride for any given distance.
The equation of line QR, which passes through points Q(0, 1) and R(2, 7), is given by the slope-intercept equation y − 1 = 3 x . This corresponds to option B. Therefore, the answer is y − 1 = 3 x .
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