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 (2)
Calculate the gradient using the formula: m = x 2 − x 1 y 2 − y 1 = 8 − 5 19 − 4 = 5 .
Substitute the gradient and one point into the equation y = m x + c : 4 = 5 ( 5 ) + c .
Solve for c : c = 4 − 25 = − 21 .
Write the final equation: y = 5 x − 21 .
Explanation
Understanding the problem 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.
Finding the gradient First, we need to find the gradient m of the line. The gradient 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 ) = ( 5 , 4 ) and ( x 2 , y 2 ) = ( 8 , 19 ) .
Calculating the gradient Substituting the coordinates of the points 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 substitute one of the points and the gradient into the equation y = m x + c to find the y-intercept c . Let's use the point ( 5 , 4 ) .
4 = 5 ( 5 ) + c 4 = 25 + c
Calculating the y-intercept Solving for c , we get: c = 4 − 25 = − 21 So, the y-intercept is c = − 21 .
Writing the equation of the line Now we can write the equation of the line in the form y = m x + c , where m = 5 and c = − 21 .
y = 5 x − 21 Thus, the equation of the 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 of a line, you can model this growth and predict its height at other weeks. This linear equation helps in understanding and predicting growth patterns in various real-life scenarios.
The equation of the straight line that passes through the points (5, 4) and (8, 19) is given by the equation y = 5x - 21. The slope of the line is 5 and the y-intercept is -21. Therefore, the final equation is in the form y = mx + c, where m = 5 and c = -21.
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