Write an equation for a line that passes through points $(2,1)$ and $(0,7)$.
A) $y=3 x+7$
B) $y=-3 x+7$
C) $y=-3 x-7$
D) $y=-7 x+3$
Answer (2)
The equation of the line that passes through the points (2, 1) and (0, 7) is given by y = -3x + 7. This matches option B from the choices provided. Therefore, the correct answer is B) y = -3x + 7.
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Calculate the slope using the formula m = x 2 − x 1 y 2 − y 1 with points ( 2 , 1 ) and ( 0 , 7 ) , which gives m = − 3 .
Identify the y-intercept as the point where x = 0 , which is ( 0 , 7 ) , so b = 7 .
Substitute the slope and y-intercept into the slope-intercept form y = m x + b to get y = − 3 x + 7 .
The equation of the line is y = − 3 x + 7 .
Explanation
Understanding the Problem We are given two points, ( 2 , 1 ) and ( 0 , 7 ) , and we need to find the equation of the line that passes through these points. The equation of a line can be written in the slope-intercept form as y = m x + b , where m is the slope and b is the y-intercept.
Finding the Slope First, we need to find the slope of the line. The slope m can be calculated using 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.
Calculating the Slope Substituting the given points ( 2 , 1 ) and ( 0 , 7 ) into the slope formula, we get: m = 0 − 2 7 − 1 = − 2 6 = − 3 So, the slope of the line is − 3 .
Finding the y-intercept Now that we have the slope, we need to find the y-intercept b . We are given the point ( 0 , 7 ) . Since the x-coordinate is 0, this point is the y-intercept. Therefore, b = 7 .
Writing the Equation of the Line Now we can write the equation of the line in slope-intercept form: y = m x + b Substituting the values we found for m and b , we get: y = − 3 x + 7
Final Answer Comparing our equation y = − 3 x + 7 with the given options, we see that it matches option B.
Examples
Imagine you are tracking the altitude of a hot air balloon over time. At 2 minutes, the balloon is at 1 mile high, and at 0 minutes, it was at 7 miles high. The equation of the line we found can help you model the balloon's descent and predict its altitude at any given time, assuming a constant rate of descent. This is a practical application of linear equations in real-world scenarios.
