A line has equation [tex]y=6 x+3[/tex]. What are the [tex]x[/tex]-intercept and [tex]y[/tex]-intercept of the line?
Answer (1)
To find the x -intercept, set y = 0 and solve for x : 0 = 6 x + 3 , which gives x = − 2 1 .
To find the y -intercept, set x = 0 and solve for y : y = 6 ( 0 ) + 3 , which gives y = 3 .
The x -intercept is − 2 1 and the y -intercept is 3 .
Therefore, the x -intercept and y -intercept of the line are − 2 1 and 3 , respectively. − 2 1 , 3
Explanation
Understanding the Problem We are given the equation of a line y = 6 x + 3 , and we need to find its x -intercept and y -intercept. The x -intercept is the point where the line crosses the x -axis, which means y = 0 . The y -intercept is the point where the line crosses the y -axis, which means x = 0 .
Finding the x-intercept To find the x -intercept, we set y = 0 in the equation y = 6 x + 3 and solve for x :
0 = 6 x + 3 Subtract 3 from both sides: − 3 = 6 x Divide both sides by 6: x = 6 − 3 = − 2 1 So the x -intercept is − 2 1 .
Finding the y-intercept To find the y -intercept, we set x = 0 in the equation y = 6 x + 3 and solve for y :
y = 6 ( 0 ) + 3 y = 0 + 3 y = 3 So the y -intercept is 3.
Final Answer Therefore, the x -intercept is − 2 1 and the y -intercept is 3.
Examples
Understanding intercepts is crucial in many real-world applications. For example, in economics, the x-intercept of a cost function can represent the break-even point, where costs equal revenue. Similarly, in physics, the y-intercept of a velocity-time graph can represent the initial velocity of an object. Knowing how to find and interpret intercepts allows us to analyze and make predictions about various phenomena.
