What is the gradient of a line with equation y = 3x + 5?

The gradient of the line with equation y = 3x + 5 is 3.

In the equation of a straight line, which is typically written in the form y = mx + c, the gradient (or slope) is represented by the coefficient of x. In this case, the equation is y = 3x + 5. Here, the coefficient of x is 3, which means the gradient of the line is 3.

The gradient of a line tells us how steep the line is. Specifically, it indicates the rate of change of y with respect to x. For every unit increase in x, the value of y increases by 3 units. This is why the gradient is sometimes referred to as the "rise over run" – it shows how much y "rises" for a given "run" along the x-axis.

Understanding the gradient is crucial when analysing the behaviour of linear functions. A positive gradient, like 3 in this case, means the line slopes upwards from left to right. Conversely, a negative gradient would indicate a line that slopes downwards. If the gradient were zero, the line would be horizontal, indicating no change in y as x changes.

In summary, the gradient of 3 in the equation y = 3x + 5 tells us that the line rises steeply, increasing by 3 units in the y-direction for every 1 unit increase in the x-direction. This concept is fundamental in GCSE Maths and helps in understanding how different linear equations behave graphically.