Which graph represents a line with a slope of $-\frac{2}{3}$ and a $y$-intercept equal to that of the line $y=\frac{2}{3} x-2$?
Answer (1)
The line has a slope of − 3 2 .
The y-intercept of the line is the same as the y-intercept of the line y = 3 2 x − 2 , which is − 2 .
The equation of the line is y = − 3 2 x − 2 .
The graph represents the line y = − 3 2 x − 2 .
Explanation
Understanding the Problem We are looking for the graph of a line with a slope of − 3 2 and a y -intercept equal to that of the line y = 3 2 x − 2 . The equation of a line is given by y = m x + c , where m is the slope and c is the y-intercept.
Finding the y-intercept First, we need to find the y-intercept of the line y = 3 2 x − 2 . The y-intercept is the value of y when x = 0 . Substituting x = 0 into the equation, we get y = 3 2 ( 0 ) − 2 = − 2 . So, the y-intercept is − 2 .
Determining the Equation of the Line Now we know that the line we are looking for has a slope of − 3 2 and a y-intercept of − 2 . Therefore, the equation of the line is y = − 3 2 x − 2 .
Finding the Graph The graph that represents the line y = − 3 2 x − 2 is a line that passes through the point ( 0 , − 2 ) and has a negative slope.
Examples
Understanding the slope and y-intercept of a line is crucial in many real-world applications. For example, if you are analyzing the cost of a taxi ride, the y-intercept could represent the initial fee, and the slope could represent the cost per mile. Similarly, in physics, the slope of a velocity-time graph represents acceleration, and the y-intercept represents the initial velocity. By understanding these concepts, you can make informed decisions and predictions in various fields.
