What is the equation of the straight line that passes through $(3,10)$ and $(7,28)$ ?
Give your answer in the form $y=m x+c$, where $m$ and $c$ are integers or fractions in their simplest forms.
Answer (1)
Calculate the gradient m using the formula m = x 2 − x 1 y 2 − y 1 , which gives m = 2 9 .
Substitute one of the points and the gradient into the equation y = m x + c to find the y-intercept c , which gives c = − 2 7 .
Write the equation of the line by substituting the values of m and c into the equation y = m x + c .
The equation of the line is y = 2 9 x − 2 7 .
Explanation
Understanding the Problem We are given two points, ( 3 , 10 ) and ( 7 , 28 ) , and we want to find the equation of the straight line that passes through these points in the form y = m x + c , where m is the gradient and c is the y-intercept.
Finding the Gradient First, we need to find the gradient m . The gradient is the change in y divided by the change in x . Using the two points, we have m = x 2 − x 1 y 2 − y 1 = 7 − 3 28 − 10 = 4 18 = 2 9 So, the gradient m = 2 9 .
Finding the Y-Intercept Now that we have the gradient, we can substitute one of the points into the equation y = m x + c to find the y-intercept c . Let's use the point ( 3 , 10 ) :
10 = 2 9 × 3 + c 10 = 2 27 + c c = 10 − 2 27 = 2 20 − 2 27 = − 2 7 So, the y-intercept c = − 2 7 .
Writing the Equation of the Line Now we can write the equation of the line by substituting the values of m and c into the equation y = m x + c :
y = 2 9 x − 2 7 Thus, the equation of the line is y = 2 9 x − 2 7 .
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 $10 for a 3-mile ride and $28 for a 7-mile ride, we can use the equation of a line to model the cost as a function of distance. This allows us to predict the cost for any given distance and helps in understanding the pricing structure of the service. Linear equations are fundamental in modeling relationships between variables in various fields, including economics, physics, and engineering.
