Find the value of $x$ so that the line through the points $A(3,5)$ and $(x, 3)$ has a slope of $\frac{3}{5}$.
Answer (1)
Use the slope formula: slope = x 2 − x 1 y 2 − y 1 .
Substitute the given values: 5 3 = x − 3 3 − 5 .
Solve the equation for x : 3 ( x − 3 ) = − 10 .
Find the value of x : x = − 3 1 .
Explanation
Understanding the Problem We are given two points, A ( 3 , 5 ) and ( x , 3 ) , and the slope of the line passing through these points is 5 3 . Our goal is to find the value of x . We will use the slope formula to set up an equation and solve for x .
Applying the Slope Formula The slope formula is given by: slope = x 2 − x 1 y 2 − y 1 where ( x 1 , y 1 ) and ( x 2 , y 2 ) are the coordinates of the two points. In our case, ( x 1 , y 1 ) = ( 3 , 5 ) and ( x 2 , y 2 ) = ( x , 3 ) , and the slope is 5 3 .
Setting up the Equation Plugging in the given values into the slope formula, we get: 5 3 = x − 3 3 − 5 5 3 = x − 3 − 2
Solving for x Now, we solve for x . Cross-multiply to get: 3 ( x − 3 ) = 5 ( − 2 ) 3 x − 9 = − 10
Isolating x Add 9 to both sides of the equation: 3 x = − 10 + 9 3 x = − 1
Finding the Value of x Divide both sides by 3: x = 3 − 1
Final Answer Therefore, the value of x is − 3 1 .
Examples
Imagine you're designing a ramp for a skateboard park. You know the starting point of the ramp and want it to end at a certain height. By using the slope formula, you can determine the horizontal distance needed to achieve the desired slope for the ramp. This ensures the ramp is neither too steep nor too gradual, making it safe and fun for skateboarders. Understanding slope is crucial in many real-world applications, from designing roads and buildings to understanding rates of change in science and economics.
