Find the equation of the line passing through the points $(6,3)$ and $(-4,3)$.
$y=[?]$
Answer (1)
Calculate the slope using the formula m = x 2 − x 1 y 2 − y 1 .
Substitute the given points ( 6 , 3 ) and ( − 4 , 3 ) into the formula to find m = 0 .
Use the point-slope form y − y 1 = m ( x − x 1 ) with point ( 6 , 3 ) and m = 0 .
Simplify the equation to get the final answer: y = 3 .
Explanation
Understanding the Problem We are given two points, ( 6 , 3 ) and ( − 4 , 3 ) , and we want to find the equation of the line that passes through them.
Finding the Slope First, we need to find the slope of the line. The slope, m , is given by the formula: m = 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 ) = ( 6 , 3 ) and ( x 2 , y 2 ) = ( − 4 , 3 ) .
Calculating the Slope Substituting the coordinates of the points into the slope formula, we get: m = − 4 − 6 3 − 3 = − 10 0 = 0 So, the slope of the line is 0.
Using the Point-Slope Form Now that we have the slope, we can use the point-slope form of a line to find the equation of the line. The point-slope form is given by: y − y 1 = m ( x − x 1 ) where ( x 1 , y 1 ) is a point on the line and m is the slope. We can use either of the given points. Let's use ( 6 , 3 ) .
Finding the Equation of the Line Substituting the slope m = 0 and the point ( 6 , 3 ) into the point-slope form, we get: y − 3 = 0 ( x − 6 ) Simplifying the equation, we have: y − 3 = 0 y = 3 Thus, the equation of the line is y = 3 .
Final Answer The equation of the line passing through the points ( 6 , 3 ) and ( − 4 , 3 ) is y = 3 .
Examples
Imagine you're designing a straight, flat road. If you know two points on the same horizontal level (like (6,3) and (-4,3)), you can easily define the road's path with the equation y = 3 . This means the road stays at a constant height of 3 units, ensuring a level surface for vehicles. Understanding linear equations helps in designing such level surfaces in various real-world applications.
