Let's analyze how the value of slope influences the graph of a linear equation!
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=2x.
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=2x
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.
