7.05 Comparing linear relationships

We've already learned how to identify linear equations, which are shown graphically as straight line graphs. Now we are going to learn how to compare the features of linear equations. Linear equations are often written in slope-intercept form which is handy because it helps us identify the slope and $y$y-intercepts of these lines as shown below.

• Slope: we may be asked which of the two linear equations has a greater/ lesser slope. This may also be referred to as the steepness of the line, the rate of change or simply increasing or decreasing faster/slower. The slope refers to how much the dependent variable changes for every time the independent variable increases by one. In context, we might be asked questions like, which carpenter charges more per hour, who ran at a quicker pace or which is the better deal?
• y-intercept: the $y$y-intercept is where the line crosses the $y$y axis. The line with the greater $y$y-intercept will be the one that crosses at a higher number on the $y$y axis. In context, we might be asked questions like, which has the higher start-up cost?
• x-intercept: this is where the line crosses the $x$x axis. Just like the y-intercept, the line with the greater $x$x-intercept will be the one that crosses at a higher number on the $x$x axis. In context, we might be asked questions like, which company will run out first or who will get home first?
• Coordinates: each pair of coordinates has an $x$x and a $y$y value $\left(x,y\right)$(x,y). We may be asked to substitute an $x$x or a $y$y value into a pair of linear equations to see which equation produces the greater/ smaller value for the other variable.

We may also be asked the compare two linear models which involve a scenario. 

Let's look through some examples that compare these different features of linear equations.

Examples

Question 1

Question 2

Question 3