Work out the equation of the straight line that passes through $(5,4)$ and $(8,19)$. 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 = 8 − 5 19 − 4 = 5 .
Substitute one of the points, say ( 5 , 4 ) , and the gradient m = 5 into the equation y = m x + c to find the y-intercept c .
Solve for c : 4 = 5 ( 5 ) + c , which gives c = − 21 .
Write the equation of the line as y = 5 x − 21 , so the final answer is y = 5 x − 21 .
Explanation
Problem Analysis We are given two points, ( 5 , 4 ) and ( 8 , 19 ) , 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.
Calculating the Gradient First, we need to calculate the gradient, m , using the formula: m = x 2 − x 1 y 2 − y 1 Substituting the given points ( 5 , 4 ) and ( 8 , 19 ) into the formula, we get: m = 8 − 5 19 − 4 = 3 15 = 5 So, the gradient of the line is m = 5 .
Finding the Y-Intercept Now that we have the gradient, we can use one of the given points to find the y-intercept, c . Let's use the point ( 5 , 4 ) . We substitute x = 5 , y = 4 , and m = 5 into the equation y = m x + c :
4 = 5 ( 5 ) + c 4 = 25 + c c = 4 − 25 = − 21 So, the y-intercept is c = − 21 .
Equation of the Line Now we have the gradient m = 5 and the y-intercept c = − 21 . We can substitute these values into the equation y = m x + c to get the equation of the line: y = 5 x − 21 Thus, the equation of the straight line is y = 5 x − 21 .
Examples
Imagine you're tracking the growth of a plant. At 5 weeks, it's 4 inches tall, and at 8 weeks, it's 19 inches tall. Using the equation y = 5 x − 21 , where y is the height and x is the number of weeks, you can predict the plant's height at any given week, assuming it grows linearly. This is a practical application of linear equations in understanding growth patterns.
