Linear equations expressed in slope-intercept form have two basic characteristics —

slope

and

y - intercept .

y = m x + b

This means that we can compare the equations by comparing their slopes and

y -

intercepts.

Slopes

Let's analyze how the value of slope influences the graph of a linear equation!

Graphs of the linear functions y=4*x, y=2*x, y=x, y=0.1*x all labeled and with different colors

Notice that the greater the slope, the steeper the line. This means that the quantity represented by $y$ increases much faster. Let's think about it using an example. Imagine two people racing at a distance of $4$ miles. The distance $y$ of the first runner is described by the equation $y = x,$ where $x$ represents time. The distance of the other runner is described by the equation $y = 2 x .$

graph of the linear functions y=2*x and y=x and graph of the constant function y=4 the functions are labeled with its function rule using a legend and the constant function is labeled as finish line

We want to know who will be first on the finish line. This means that we want to find out who reaches the finish line in shorter time, with a smaller value of $x .$ As we can see, it is $the$ $second$ $runner,$ with the greater slope.

Equation	Slope
$y = x$	$1$
$y = 2 x$	$2$

The line with a greater slope always will be steeper. Therefore, knowing the slopes of the linear equations, we know who is faster and who will win the race, even without looking at the graph.

$y -$ intercepts

To compare linear equations we can also analyze the value of $y -$ intercept. Let's think about lines with the same slope and different $y -$ intercepts.

All lines are parallel and the lines with greater $y -$ intercepts are above the lines with smaller $y -$ intercepts. Let's again think about the racing example.

In this case, the runners start simultaneously, but $one$ $of$ $them$ starts $2$ miles ahead. When they run with the same speed, and therefore their distance equations have the same slope, the first person on the finish line will be $the$ $runner$ with greater $y -$ intercept. This is the person who starts $2$ miles ahead.

Conclusions

When comparing linear equations, we can analyze the connections between slopes and between $y -$ intercepts. The greater the slope, the steeper the line. The greater the $y -$ intercept, the higher the line is located. We can use this information to solve problems, for example to consider which line will first reach a specified value.

Slopes

y-intercepts

Conclusions

$y -$ intercepts

Exercise