Find the equation of the line passing through the points $(3,3)$ and $(4,5)$.
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Answer (1)
Calculate the slope m using the formula m = x 2 − x 1 y 2 − y 1 , which gives m = 4 − 3 5 − 3 = 2 .
Substitute the slope and one of the points (e.g., ( 3 , 3 ) ) into the equation y = m x + b to find the y-intercept b .
Solve for b : 3 = 2 ( 3 ) + b , which gives b = − 3 .
Write the final equation of the line as y = 2 x − 3 , so the answer is y = 2 x − 3 .
Explanation
Understanding the Problem We are given two points, ( 3 , 3 ) and ( 4 , 5 ) , and we want to find the equation of the line that passes through them. The equation of a line can be written in the form y = m x + b , where m is the slope and b is the y-intercept.
Calculating the Slope First, we need to calculate the slope m of the line. The slope is defined as the change in y divided by the change in x . Using the given points ( x 1 , y 1 ) = ( 3 , 3 ) and ( x 2 , y 2 ) = ( 4 , 5 ) , we have: m = x 2 − x 1 y 2 − y 1 = 4 − 3 5 − 3 = 1 2 = 2
Finding the Y-intercept Now that we have the slope m = 2 , we can substitute one of the points into the equation y = m x + b to solve for the y-intercept b . Let's use the point ( 3 , 3 ) :
3 = 2 ( 3 ) + b 3 = 6 + b b = 3 − 6 = − 3
Writing the Equation of the Line Now we have the slope m = 2 and the y-intercept b = − 3 . We can write the equation of the line as: y = 2 x − 3
Final Answer Therefore, the equation of the line passing through the points ( 3 , 3 ) and ( 4 , 5 ) is y = 2 x − 3 .
Examples
Imagine you're tracking the growth of a plant. At 3 weeks, it's 3 inches tall, and at 4 weeks, it's 5 inches tall. Finding the equation of the line helps you predict its height at other weeks, assuming it grows at a constant rate. This is a linear relationship, where each week the plant grows by a consistent amount. Understanding linear equations allows you to make predictions based on observed data in various real-world scenarios.
