Finding the Gradient of a Straight Line

Gradient? Isn’t that like a colour thing?


In maths the gradient has a slightly different meaning.


The gradient of a straight line shows us how steep the line is. Think of the mountain side.


In worded form, the gradient is rise over run.







Types of Gradients


There are two types of gradients; negative and positive.


Negative gradients go from up to down. Like when you drop your ice cream, your mood goes from up to down. Now you’re feeling negative :(











Positive gradients go from down to up. Like when you get another ice cream your mood goes up and you’re back to feeling positive :)



Finding the Gradient in a Linear Equation


A linear equation shows the equation of a straight line.









Here we can see that m = gradient and c = y intercept.

Note: the y intercept is the point on the y axis where the line crosses.


So, if we have the equation of a straight line, we can simply find m and there’s our gradient.

Note: m is the value that is multiplied by x


Example 1: Find the gradient of this equation.





We can see that this equation has the same format as our linear equation. We just need to locate m.







This means that ½ is our gradient.

So, to answer the question, the gradient of this line is ½.


Now what if our equation doesn’t have the same format as our linear equation?


Don’t stress! All we have to do is rearrange it a bit by transposing! If you need a refresher course on transposing, click here for our blog post.



Example 2:





Here, our equation is not in the same format as the linear equation. Let’s start moving things around to try get that y by itself.


First, we’re going to move the x to the other side of the equals sign.

Note: x is positive so to move it over, we change the sign to a negative.







This gives us




Now, to get y by itself, we need to move that 3.

Note: the 3 is multiplied so to move it over, we need to divide.






This give us






Next, we simplify.






So our final answer is





Okay, but wait, which one is m?


Well, it's the number in front of the x so in this case it would be -⅓.






Therefore, the gradient is -⅓



Finding the Gradient Using 2 Points


For these types of question we need to use a formula






Remember, m is the gradient.


The x and y just refer to the points given in the question.


So, x1 is the x coordinate of the first point and y1 is the y coordinate of the first point.


x2 is the x coordinate of the second point and y2 is the second coordinate of the second point.


Let’s look at an example.


Example 1: Find the gradient between these two points




Step 1: Label the coordinates.





Step 2: Write out your gradient formula.






Step 3: Plug in your coordinates into the formula.






Step 4: As always, we have to simplify!






Step 5: Write your gradient.



That’s it! Super simple!

Finding the gradient seems like a daunting task but follow these easy steps and you’ll be a pro in no time!


For more tips and tricks check out our other blogs on our website. For more personalised advice, try out one of our classes! You can find more information on our website including some tasty deals!


Study smarter, not harder, with SAC!


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