What is the equation of the line with an $x$-intercept of -2 and a $y$-intercept of 1?
A. $y=\frac{1}{2} x+1$
B. $y=2 x+1$
C. $y=-\frac{1}{2} x+1$
Answer (1)
Calculate the slope using the formula m = x 2 − x 1 y 2 − y 1 with points ( − 2 , 0 ) and ( 0 , 1 ) , which gives m = 2 1 .
Use the slope-intercept form of a line, y = m x + b , where m is the slope and b is the y-intercept.
Substitute the calculated slope m = 2 1 and the given y-intercept b = 1 into the equation.
The equation of the line is y = 2 1 x + 1 .
Explanation
Understanding the Problem We are given that the line has an x -intercept of -2 and a y -intercept of 1. This means the line passes through the points ( − 2 , 0 ) and ( 0 , 1 ) . Our goal is to find the equation of this line.
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 two points on the line. In our case, we have ( x 1 , y 1 ) = ( − 2 , 0 ) and ( x 2 , y 2 ) = ( 0 , 1 ) .
Calculating the Slope Substituting the coordinates of the points into the slope formula, we get: m = 0 − ( − 2 ) 1 − 0 = 2 1 So, the slope of the line is 2 1 .
Using Slope-Intercept Form Now we can write the equation of the line in slope-intercept form, which is: y = m x + b where m is the slope and b is the y -intercept. We already know that the y -intercept is 1, so b = 1 .
Finding the Equation of the Line Substituting the slope m = 2 1 and the y -intercept b = 1 into the equation, we get: y = 2 1 x + 1 Thus, the equation of the line is y = 2 1 x + 1 .
Examples
Imagine you're designing a ramp for a skateboard park. You want the ramp to start at a certain point on the ground (the x-intercept) and reach a certain height (the y-intercept). By finding the equation of the line that represents the ramp, you can determine the slope and ensure the ramp is safe and fun to ride. This problem demonstrates how linear equations can be used to model real-world scenarios involving slopes and intercepts.
