Which equation represents the line that passes through points $(0,6)$ and $(2,0)$?
A. $y=-\frac{1}{3} x+2$
B. $y=-\frac{1}{3} x+6$
C. $y=-3 x+2$
D. $y=-3 x+6$
Answer (2)
The equation of the line passing through the points (0, 6) and (2, 0) is found by calculating the slope and using the slope-intercept form. The slope is -3, and the y-intercept is 6, leading to the equation y = − 3 x + 6 . Therefore, the correct choice is option D.
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Calculate the slope using the formula: m = x 2 − x 1 y 2 − y 1 = 2 − 0 0 − 6 = − 3 .
Use the slope-intercept form of a line: y = m x + b .
Substitute the point ( 0 , 6 ) to find the y-intercept: 6 = − 3 ( 0 ) + b , so b = 6 .
Write the equation of the line: y = − 3 x + 6 , so the final answer is y = − 3 x + 6 .
Explanation
Understanding the Problem We are given two points, ( 0 , 6 ) and ( 2 , 0 ) , and we need to find the equation of the line that passes through these points.
Finding the Slope First, we need to find 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.
Calculating the Slope Using the given points ( 0 , 6 ) and ( 2 , 0 ) , we have: m = 2 − 0 0 − 6 = 2 − 6 = − 3 So, the slope of the line is − 3 .
Using Slope-Intercept Form Now we know that the equation of the line is in the form y = m x + b , where m is the slope and b is the y-intercept. We already found that m = − 3 , so the equation is y = − 3 x + b .
Finding the Y-Intercept Since the line passes through the point ( 0 , 6 ) , we can substitute x = 0 and y = 6 into the equation to find b :
6 = − 3 ( 0 ) + b 6 = 0 + b b = 6 So, the y-intercept is 6 .
Writing the Equation Now we have the slope m = − 3 and the y-intercept b = 6 . We can write the equation of the line as: y = − 3 x + 6 Comparing this equation with the given options, we see that it matches the fourth option.
Final Answer Therefore, the equation of the line that passes through the points ( 0 , 6 ) and ( 2 , 0 ) is y = − 3 x + 6 .
Examples
Linear equations are used in various real-life applications, such as determining the cost of a taxi ride based on the distance traveled. For example, a taxi company might charge a fixed fee plus a per-mile rate. If the fixed fee is $6 and the per-mile rate is 3 , t h ee q u a t i o n re p rese n t in g t h e t o t a l cos t y f or x mi l es i s y = 3x + 6$. This is a linear equation, and understanding how to find such equations helps in everyday financial planning and decision-making.
