Find the equation (in terms of x) of the line through the points (-1,-5) and (5,3).
Answer (1)
Calculate the slope using the formula: m = x 2 − x 1 y 2 − y 1 = 5 − ( − 1 ) 3 − ( − 5 ) = 3 4 .
Use the point-slope form with point (-1, -5): y − ( − 5 ) = 3 4 ( x − ( − 1 )) .
Simplify the equation: y + 5 = 3 4 ( x + 1 ) .
Solve for y to get the equation of the line: y = 3 4 x − 3 11 .
y = 3 4 x − 3 11
Explanation
Problem Analysis We are given two points, (-1, -5) and (5, 3), and we want to find the equation of the line that passes through these points. The equation should be in the form y = mx + 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 the change in y divided by the change in x. Using the formula: m = x 2 − x 1 y 2 − y 1 where ( x 1 , y 1 ) = ( − 1 , − 5 ) and ( x 2 , y 2 ) = ( 5 , 3 ) .
Substituting the values, we get: m = 5 − ( − 1 ) 3 − ( − 5 ) = 5 + 1 3 + 5 = 6 8 = 3 4 So, the slope of the line is 3 4 .
Using Point-Slope Form Now that we have the slope, we can use the point-slope form of a linear equation to find the equation of the line. The point-slope form is: y − y 1 = m ( x − x 1 ) We can use either of the given points. Let's use the point (-1, -5). Substituting the values, we get: y − ( − 5 ) = 3 4 ( x − ( − 1 )) y + 5 = 3 4 ( x + 1 ) Now, we solve for y to get the equation in the form y = mx + b:
Finding the Equation y + 5 = 3 4 x + 3 4 y = 3 4 x + 3 4 − 5 To combine the constants, we need a common denominator. Since 5 can be written as 3 15 , we have: y = 3 4 x + 3 4 − 3 15 y = 3 4 x − 3 11 So, the equation of the line is y = 3 4 x − 3 11 .
Final Answer The equation of the line that passes through the points (-1, -5) and (5, 3) is: y = 3 4 x − 3 11
Examples
Understanding linear equations is crucial in many real-world applications. For instance, consider a taxi service that charges a fixed fee plus a per-mile rate. If the taxi charges a $3 fixed fee and $2 per mile, the total cost can be modeled by a linear equation. Similarly, in physics, the relationship between distance, speed, and time for an object moving at a constant speed can be represented by a linear equation. These equations help us predict outcomes, optimize processes, and make informed decisions in various fields.
