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.
Checking Equation 1 Let's check 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 .
Checking Equation 2 Now, let's check 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 .
Checking Equation 3 Next, let's check 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 .
Checking Equation 4 Finally, let's check 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 .
Final Answer 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 instance, if you're designing a road that needs to pass through a specific location, you can use this method to verify if the road's path (represented by an equation) will indeed go through that location (represented by a point). This ensures that your design meets the required constraints. Similarly, in physics, if you're tracking the trajectory of a projectile, you can check if its path passes through a certain point in space using the same principle.
