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The slope of a line

The slope of a line

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Two lines with the same k-value will...

The slope of a line

​If you've mastered proportional change and constance, it's time to move on and take a closer look at how calculate a slope of a line. To calculate a slope of a line, we need two points on a line. We choose A an​​d B on this line. The slope tells us how far up or down a line goes for every step forward. The two dotted lines show the vertical and horizontal change.

In other words, how much Y changes when X is changed by a certain amount. The length of the vertical dotted line is equal to the difference in Y values between the points A and B. And the length of the horizontal line is equal to the difference in X values between the two points. The slope is the quotient of these two lengths. If there is a larger difference between the Y values, you get a larger quotient and a steeper slope.

If there is a larger difference between the X values you get a smaller quotient and a less steep slope. If the difference between the X values equals the difference between the Y values the slopes equals 1 which means a slope of 45 degrees. ​ ​Now we are going to do some math to find the numerical value of the slope. To get the difference in Y values take the Y value of point A minus the Y value of point B, six minus two. Divide that by the difference between the X values. The X value of point A minus the X value of point B, three minus one. It doesn't matter which point you start with as long as you take them in the same order for both the denominator and numerator.

It also doesn't matter which two points you choose. Since we only want to get the relative difference between the vertical and horizontal changes, this works with any two points on the line. Do the subtractions and division and you get two. The line that goes through points A and B has the slope of two. We usually use the letter K to represent a slope or a gradient.

Let's write that K equals two. ​​ This means that for every one step forward the line goes two steps upwards. ​ ​Look at this line, it slopes the other way, downwards. Let's calculate the line slope and see what happens. Use the formula and let C be X1 and Y1, and D be X2 and Y2. One minus two is minus one, and four minus one is three. If both the numerator and the denominator were negative numbers the quotient would be positive.

But here, only one of them is negative and we get the quotient of minus one-third. The line slope is a negative number, and the line slopes downwards. And downwards is exactly what a negative slope means. ​ ​For every step to the right, the line goes a third of a step down. A negative slope means a line sloping down. Can you see that these two lines have the same slope?

The new line does not go through points C and D, but the slope is the same. When two lines have the same slope, they are parallel. This means that they never cross. Two straight lines can either have exactly one point in common or have the same slope, but they can never share both the slope and only one point. ​ ​The slope of a line is a measure of how steep it is. We usually represent this with the letter K.

If X represents time, then the slope of a line describes the rate of change. A steeper line means a greater slope and a greater value of K. ​ ​To calculate the slope of a line, you only need to know two points on the line: by subtracting one point's coordinates from the others and dividing the answers, you get the relative difference between the points. Lines with positive slopes go upwards to the right, and lines with negative slopes go down. Two lines that are parallel have the same slope, but never cross each other.