Grade 9 Maths Book An Easy Way to Learn Maths

In our everyday environment we notice many surfaces that are slanted rather than perfectly flat. Think of the gentle incline of a ramp at your school, or the varying steepness of a hill on a familiar route. These everyday features are examples of what we call a \(\textbf{gradient}\text{.}\)

Instead of focusing on the mathematics immediately, it is important to understand that the gradient describes how steep a path is. It tells us how quickly something rises or falls over a certain distance.

In practical terms, this concept is useful in planning safe ramps, roads, or even understanding the effort required to climb a hill.

Key Takeaway:

The gradient gives a numerical value to the concept of steepness, allowing us to compare different slopes and understand their practical implications for safety, accessibility, and design.

In many situations, we are given two points on a straight path and want to determine how steep that path is. These two points may represent physical locations, price changes, distances over time, or any relationship that forms a straight line.

To calculate the gradient between any two points on a straight line, we compare how much the value changes vertically (rise) with how much it changes horizontally (run).

Gradients are useful in describing real situations such as speed, price change, or slope of land.

Key Takeaway:

The gradient between two points shows how steep the line is. It is found by dividing the change in \(y\) values (rise) by the change in \(x\) values (run).

Once you have learned how to calculate the gradient between two points, the next step is to use those two points to determine the unique equation of the line that connects them.

In this section, we will introduce the idea that every straight line in the coordinate plane can be defined by its \(\textbf{gradient}\) and a point it passes through. This leads us to the \(\textbf{point-slope form}\) of a line, which can later be rearranged into the slope-intercept form:

\(y = mx + c \)

where \(m\) is the gradient and \(c\) is the y-intercept.

Today, we will combine your earlier work with a new process: determining the line’s equation directly from two known points.

Key Takeaway:

By determining the gradient between two points and substituting one of the points into the point-slope equation, we can derive the unique linear equation that passes through both points. This equation can then be rearranged into the form \(y = mx + c\text{.}\)

In some cases, we may be told the \(\textbf{gradient (m)}\) of a line and given only one point that lies on it. Using this information, we can still find the full equation of the line.

To do this, we use the \(\textbf{point-slope form}\) of a linear equation:

\(y - y_1 = m(x - x_1) \)

where \((x_1, y_1)\) is the known point and \(m\) is the gradient. We can rearrange this into the familiar slope-intercept form: \(y = mx + c \text{.}\)

Key Takeaway:

When given a gradient and one point on a line, you can use the point-slope form to find the full equation of the line. This method is efficient and avoids the need for a second point.

When sketching or interpreting a linear graph, it is helpful to know where the line crosses the axes. These points are called the \(\textbf{intercepts}\text{.}\)

Knowing both intercepts helps you to sketch a line quickly and accurately.

Key Takeaway:

The y-intercept is found by setting \(x = 0\text{.}\) The x-intercept is found by setting \(y = 0\text{.}\) These two points are enough to sketch a linear graph accurately.

Linear equations are not just for the classroom, they appear all around us in daily life. Any situation that involves a fixed starting value and a constant rate of change can be modelled using a straight-line equation of the form \(y = mx + c\text{.}\)

Understanding these applications can help us make predictions, compare situations, and make informed decisions.

Key Takeaway:

Everyday life situations involving steady increase or decrease can be modelled using linear equations. This makes it easier to predict, budget, or compare.