Finding the Equation of a Straight Line

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Juliet Mitchell
5 years ago
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1 Finding the Equation of a Straight Line You should have, before now, ome aross the equation of a straight line, perhaps at shool. Engineers use this equation to help determine how one quantity is related to another this an be for many reasons as you will find out throughout this ourse. Mathematially the equation of a straight line is given as Where: y mx or y axb x is the x o-ordinate (x, y) y is the y o-ordinate (x, y) m or a is the slope of the line [rise run ] or b is where the line rosses the y axes [also known as the y interept] Chek these out on the graph in Figure 1. Figure 1: Simple Plot If we know all of the above information we an use this to find the atual mathematial relationship between the x and y values. That is, ultimately, if we know the value of x we an alulate y. Why would we want to do this? Well, in many modern omputer ontrolled systems a sensors output is used to determine a value for example the position of a printer head the output of the position sensor is a voltage and the omputer uses this to determine the displaement of the printhead in mm. Let s see how it works. MjD Aug of 7

2 Chek out Figure 2. Here we an see two plots, one whih appears to slope up the way, plot height inreases as the x value inreases and the other appears to slope down, plot height dereases as the x value inreases. We will hek out the straight line equation for both of these plots and see how the straight line equation desribes eah. The straight line equation: for the rest of this hand-out we will use y mx First we identify and selet two o-ordinate pairs from one of the plots that best desribes the plot. Let s hoose values from the line that slopes upwards first. Looking at the plot from the origin (0,0) we an see that the left hand end of the line is sitting at the point x1 = 1 and y1 = 2 also written (1,2) while the other end of the line is at x2 = 8 and y2 = 8 or (8,8). We will use these as our two sets of o-ordinates. Figure 2: Two different plots. x1 = 1 x2 = 8 y1 = 2 y2 = 8 We an now use the o-ordinate pair to find the slope of the line m. The slope of the line is a measure of how steep it is and is alulated by finding out how high [the rise] it moves over what distane [the run]. To find m we arry out the following alulation: rise m = run We an determine the rise from the o-ordinates; it is the differene in the y oordinates or y2 y1 = 8 2 MjD Aug of 7

3 Whilst for the run; it is the differene in the x o-ordinates or x 2 x 1 ; in this ase 8 1 Plaing this in the equation for m: 8 2 m = = 7 = This allows us to re-write our partially ompleted straight line equation as: y = x We now have to determine the (y interept) or. In order to find we first have to re-arrange the straight line equation so that is on the left of the equals sign. y mx First we need to move the mx term over to the left of the equation in order to do this we subtrat mx from both sides [we have to subtrat from both sides in order to keep the equation balaned]. y mx = mx mx+ We an re-write this as: y mx = + but this ould also be written as y mx = or = y mx We an now use this equation and our previous o-ordinate pairs to find. We know the values of some of the points on the line that makes up our plot; let s use the first o-ordinate pair x1 and y1 by substituting these values into our new equation we an alulate. [N.B remember m has a value of ] x = x1 = 1 y = y1 = 2 m = = y mx = = = We an now write out the full equation for the straight line graph: y = x How do we know that the equation is orret? We use the other o-ordinate pair to hek it. We substitute x2 in for x and alulate through the answer should be y2. MjD Aug of 7

4 x2 = 8 y2 = 8 y = x As expeted. y = = = 8 MjD Aug of 7

5 The equation: y = mx+ For the seond plot. Again we identify and selet two o-ordinate pairs from the plot. Looking at the plot from the origin (0,0) we an see that the left hand end of the line is sitting at the point x1 = 1 and y1 = 6 also written (1,6) while the other end of the line is at x2 = 5 and y2 = 1 or (5,1). We will use these as our two sets of o-ordinates. Figure 3: Two different plots. x1 = 1 x2 = 5 y1 = 6 y2 = 1 We an now use the o-ordinate pair to find the slope of the line m. The slope of the line is a measure of how steep it is and is alulated by finding out how high [rise] it moves in what distane [run]. To find m we arry out the following alulation: rise m run We an determine the rise from the o-ordinates it is the differene in the y oordinates or y2 y1 = 1 6 Whilst the run is the differene in the x o-ordinates or x2 x1 in this ase 5 1 Plaing this in the equation for m: 1 6 m = = 4 = 1.25 This allows us to re-write our partially ompleted straight line equation as: y = x We now have to determine the (y interept) or. NB a negative sign in front of m just shows us that the plot is sloping downwards. MjD Aug of 7

6 In order to find we first have to re-arrange the straight line equation so that is on the left of the equals sign. y = mx+ First we need to move the mx term over to the left of the equation in order to do this we subtrat mx from both sides [we have to subtrat from both sides in order to keep the equation balaned]. y mx = mx mx+ We an re-write this as: y mx = + but this ould also be written as y mx = or = y mx We an now use this equation and our previous o-ordinate pairs to find. We know the values of some of the points on the line that makes up our plot; let s use the first o-ordinate pair x1 and y1 by substituting these values into our new equation we an alulate. [N.B remember m has a value of ] x = x1 = 6 y = y1 = 1 m = = y mx = 6 ( 1.251) = = We an now write out the full equation for the straight line graph: y = x How do we know that the equation is orret? We use the other o-ordinate pair to hek it. We substitute x2 in for x and alulate through the answer y should be y2. x2 = 5 y2 = 1 As expeted. y = x 1= = = 1 NB a negative sign in front of m just shows us that the plot is sloping downwards. MjD Aug of 7

7 Knowing the equation for the straight line allows us to alulate the y value for any given x value. We an also extend our equation manipulation to let us find where the line rosses the x axis or the x interept. If we are looking to find where the graph line rosses the x axis then at that point the y value must be equal to zero y=0: If we now arry this knowledge bak to our equation of a straight line we an see: x 1.25 x 5.80 y = x = x x( 1.25) If we now examine the graph we an see this is what would be expeted. MjD Aug of 7

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