Select the equations that contain the point $(-3,5)$.
$y=-3 x+5$
$y=-3 x-4$
$y=-x+2$
$y=-x+5$
Answer (1)
Substitute the point ( − 3 , 5 ) into each equation.
Check if the equation holds true after the substitution.
The equation y = − 3 x − 4 holds true: 5 = − 3 ( − 3 ) − 4 = 9 − 4 = 5 .
The equation y = − x + 2 holds true: 5 = − ( − 3 ) + 2 = 3 + 2 = 5 .
The equations that contain the point ( − 3 , 5 ) are y = − 3 x − 4 and y = − x + 2 .
Explanation
Problem Analysis We are given the point ( − 3 , 5 ) and four equations. We need to determine which of these equations are satisfied by the given point. To do this, we will substitute x = − 3 and y = 5 into each equation and check if the equation holds true.
Testing Equation 1 Let's test the first equation: y = − 3 x + 5 . Substituting x = − 3 and y = 5 , we get: 5 = − 3 ( − 3 ) + 5 5 = 9 + 5 5 = 14 This is false, so the point ( − 3 , 5 ) does not lie on the line y = − 3 x + 5 .
Testing Equation 2 Now, let's test the second equation: y = − 3 x − 4 . Substituting x = − 3 and y = 5 , we get: 5 = − 3 ( − 3 ) − 4 5 = 9 − 4 5 = 5 This is true, so the point ( − 3 , 5 ) lies on the line y = − 3 x − 4 .
Testing Equation 3 Next, let's test the third equation: y = − x + 2 . Substituting x = − 3 and y = 5 , we get: 5 = − ( − 3 ) + 2 5 = 3 + 2 5 = 5 This is true, so the point ( − 3 , 5 ) lies on the line y = − x + 2 .
Testing Equation 4 Finally, let's test the fourth equation: y = − x + 5 . Substituting x = − 3 and y = 5 , we get: 5 = − ( − 3 ) + 5 5 = 3 + 5 5 = 8 This is false, so the point ( − 3 , 5 ) does not lie on the line y = − x + 5 .
Conclusion Therefore, the equations that contain the point ( − 3 , 5 ) are y = − 3 x − 4 and y = − x + 2 .
Examples
In coordinate geometry, determining whether a point lies on a given line is a fundamental concept. For example, if you're designing a straight-line route for a delivery service, you need to ensure that certain key locations (points) fall directly on that route (line). Similarly, in computer graphics, this principle is used to verify if a pixel (point) is part of a rendered line or shape. This concept is also crucial in fields like physics, where you might want to confirm if an object's trajectory (line) passes through a specific sensor's location (point).
