Linear relationships are important ideas in math, especially in algebra. They are shown as straight lines on a graph. You can describe these relationships with an equation that looks like this:

$$y = mx + c$$

Here’s what those letters mean:

• $y$ is the outcome or dependent variable.
• $x$ is the input or independent variable.
• $m$ is the slope of the line, showing how steep it is.
• $c$ is the y-intercept, which tells us where the line crosses the y-axis.

Features of Linear Relationships

1. Constant Rate of Change: In a linear relationship, the way $y$ changes when $x$ changes is steady. This means if $x$ goes up by a certain amount, $y$ will go up (or down) by a specific amount too.

2. Graph Shape: When you graph a linear relationship, it will always make a straight line. The direction of this line depends on $m$:

   • If $m$ is positive, the line goes up as you move to the right.
   • If $m$ is negative, the line goes down.
   • If $m$ is zero, the line is flat.

3. Y-Intercept: The value of $c$ tells us where the line crosses the y-axis (the line that goes up and down). For example, if $c = 2$, the line crosses the y-axis at the point (0, 2).

How to Graph Linear Relationships

To graph a linear equation, you can follow these steps:

1. Find the Slope and Y-Intercept:

   • From the equation $y = mx + c$, find $m$ and $c$. For example, in $y = 2x + 3$, the slope $m$ is 2 and the y-intercept $c$ is 3.

2. Plot the Y-Intercept: Begin by plotting the y-intercept on the graph. If $c = 3$, mark the point (0, 3) on the y-axis.

3. Use the Slope: The slope $m$ can be written as a fraction. For example, if $m = 2$, this can be seen as $\frac{2}{1}$. This means you move up 2 units for every 1 unit you move to the right. From (0, 3), go up 2 units and right 1 unit to get to (1, 5). Plot this point.

4. Draw the Line: Connect the points you plotted with a straight line. Make sure to extend the line in both directions and add arrows to show that it keeps going.

Example

Let’s look at the linear equation $y = -\frac{1}{2}x + 4$.

1. Identify Components:

   • The slope $m = -\frac{1}{2}$ (this means the line goes down).
   • The y-intercept $c = 4$ (the line crosses the y-axis at (0, 4)).

2. Plot and Use the Slope:

   • Start at (0, 4).
   • From (0, 4), go down 1 unit and right 2 units to find another point at (2, 3).
   • Plot the point (2, 3).

3. Connect Points: Draw a straight line through the points (0, 4) and (2, 3). Extend the line with arrows on both ends.

Conclusion

Understanding linear relationships helps us make sense of real-world situations. This is an important part of Year 7 math. Knowing how to graph these relationships gives students valuable skills for working with data in many subjects, not just math!