What is the $x$-intercept of the graph of the equation $y=-2x+6$ in the $xy$-plane?
Answer (1)
To find the x -intercept, set y = 0 in the equation.
Solve the equation 0 = − 2 x + 6 for x .
Divide both sides by 2: x = 2 6 = 3 .
The x -intercept is ( 3 , 0 ) .
Explanation
Understanding the Problem The problem asks for the x -intercept of the graph of the equation y = − 2 x + 6 . The x -intercept is the point where the graph intersects the x -axis. This occurs when y = 0 .
Setting y = 0 To find the x -intercept, we set y = 0 in the equation y = − 2 x + 6 :
0 = − 2 x + 6
Solving for x Now, we solve for x :
2 x = 6 x = 2 6 x = 3
Finding the x-intercept The x -intercept is the point where y = 0 , so the x -intercept is ( 3 , 0 ) .
Examples
Understanding x-intercepts is crucial in many real-world applications. For example, if you're analyzing a business's profit and loss, the x-intercept represents the point at which the business breaks even – neither making a profit nor incurring a loss. Similarly, in physics, if you're studying the motion of an object, the x-intercept on a velocity-time graph could represent when the object changes direction. Knowing how to find and interpret x-intercepts allows you to make informed decisions and predictions in various fields.
