Determine whether the point with coordinates $(1,6)$ lies on the line with equation $y=2x+6$.
Answer (2)
Substitute the x-coordinate of the point into the line's equation.
Calculate the corresponding y-value: y = 2 ( 1 ) + 6 = 8 .
Compare the calculated y-value with the point's y-coordinate.
Since 8 e q 6 , the point does not lie on the line. The point does not lie on the line F a l se .
Explanation
Problem Analysis We are given a point with coordinates ( 1 , 6 ) and a line with the equation y = 2 x + 6 . We want to determine if the point lies on the line. To do this, we will substitute the x -coordinate of the point into the equation of the line and see if the resulting y -value matches the y -coordinate of the point.
Substitution Substitute x = 1 into the equation y = 2 x + 6 :
y = 2 ( 1 ) + 6
Calculation Calculate the value of y :
y = 2 + 6 = 8
Comparison and Conclusion The calculated y -value is 8, but the y -coordinate of the given point is 6. Since 8 e q 6 , the point ( 1 , 6 ) does not lie on the line y = 2 x + 6 .
Examples
In architecture, determining if a point lies on a line is crucial for designing structures. For example, an architect might need to verify if a support beam with coordinates (x, y) aligns with a design blueprint represented by a linear equation. By substituting the coordinates into the equation, they can confirm the beam's placement, ensuring structural integrity and aesthetic precision. This principle extends to various fields, including computer graphics, navigation, and engineering, where accurate spatial relationships are essential for success.
The point ( 1 , 6 ) does not lie on the line described by the equation y = 2 x + 6 , as substituting x = 1 gives a calculated y-value of 8, which differs from the point's y-coordinate of 6.
;
