Find the $y$-intercept and $x$-intercept of the line.
$-x+2 y=2$
Answer (2)
To find the y -intercept, set x = 0 and solve for y , which gives y = 1 .
To find the x -intercept, set y = 0 and solve for x , which gives x = − 2 .
The y -intercept is 1 and the x -intercept is -2.
Therefore, the y -intercept is 1 and the x -intercept is − 2 .
Explanation
Understanding the Problem We are given the equation of a line: − x + 2 y = 2 . Our goal is to find the x -intercept and the y -intercept of this line.
Finding the y-intercept To find the y -intercept, we set x = 0 in the equation and solve for y . Substituting x = 0 into the equation − x + 2 y = 2 , we get − ( 0 ) + 2 y = 2 , which simplifies to 2 y = 2 .
Calculating the y-intercept Dividing both sides of the equation 2 y = 2 by 2, we find y = 2 2 = 1 . Therefore, the y -intercept is 1, and the coordinate is ( 0 , 1 ) .
Finding the x-intercept To find the x -intercept, we set y = 0 in the equation and solve for x . Substituting y = 0 into the equation − x + 2 y = 2 , we get − x + 2 ( 0 ) = 2 , which simplifies to − x = 2 .
Calculating the x-intercept Multiplying both sides of the equation − x = 2 by -1, we find x = − 2 . Therefore, the x -intercept is -2, and the coordinate is ( − 2 , 0 ) .
Final Answer In summary, the y -intercept is 1 (the point ( 0 , 1 ) ), and the x -intercept is -2 (the point ( − 2 , 0 ) ).
Examples
Understanding intercepts is crucial in various real-world applications. For instance, in economics, the y -intercept of a cost function represents the fixed costs, while the x -intercept of a demand function indicates the saturation point where demand becomes zero. Similarly, in physics, intercepts can represent initial conditions or equilibrium points. Knowing how to find intercepts allows us to interpret and analyze linear relationships in many different fields.
The y -intercept of the line − x + 2 y = 2 is 1, found at the point ( 0 , 1 ) , and the x -intercept is -2, found at the point ( − 2 , 0 ) .
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