We want to explain how slopes and $y -$ intercepts of the equations in a system of linear equations relate to the number of solutions of the system. To do so, we will examine different cases of slopes and $y -$ intercepts of equations one at a time. We will also determine how many intersection points they have because it shows the number of solutions of the system.

Case I: Different Slopes, Different $y -$ intercepts

Suppose we have two linear equations with different

s l o p e s

and

y - intercepts,

and write them in slope-intercept form.

{y = 2 x - 4 y = - 3 x + 3 \Rightarrow {y = 2 x + (- 4) y = - 3 x + 3

Let's now look at their graphs and see how many points of intersection they have.

Different Slopes, Different y-intercepts

As can be seen on the graph, the equations have only one point of intersection. This implies that this system of equations has one solution.

Case II: Different Slopes, Same $y -$ intercepts

Assume that we have two linear equations with different

s l o p e s

but the same

y - intercepts .

We will first write them in slope-intercept form.

{y = x + 4 y = - 3 x + 4 \Rightarrow {y = 1 x + 4 y = - 3 x + 4

Next, we will look at their graphs and see how many points of intersection they have.

The graph shows that the equations have only one point of intersection — in this case their $y -$ intercept. Therefore, this system of equations has one solution.

Case III: Same Slopes, Different $y -$ intercepts

Consider two linear equations with the same

s l o p e s

but different

y - intercepts,

and then write them in slope-intercept form.

{y = x + 1 y = x - 2 \Rightarrow {y = 1 x + 1 y = 1 x + (- 2)

We will now look at the graphs of these equations and identify how many points of intersection they have.

Note that the equations do not intersect. Therefore, this system of equations has no solution.

Case IV: Same Slopes, Same $y -$ intercepts

Take two linear equations with the same

s l o p e s

and

y - intercepts,

and write them in slope-intercept form.

{y = x + 1 y = 1 + x \Rightarrow {y = 1 x + 1 y = 1 x + 1

Let's now examine the graphs of these equations and determine their intersection points.

Note that the graphs of the equations coincide. Therefore, this system of equations has infinitely many solutions.

Conclusion

The following table summarizes how the slopes and $y -$ intercepts are related to the number of solutions of a system of linear equations.

Slopes	$y -$ intercepts	Number of Solutions
Different	Different or Same	One Solution
Same	Different	No Solution
Same	Same	Infinitely Many Solutions

Case I: Different Slopes, Different y-intercepts

Case II: Different Slopes, Same y-intercepts

Case III: Same Slopes, Different y-intercepts

Case IV: Same Slopes, Same y-intercepts

Conclusion

Case I: Different Slopes, Different $y -$ intercepts

Case II: Different Slopes, Same $y -$ intercepts

Case III: Same Slopes, Different $y -$ intercepts

Case IV: Same Slopes, Same $y -$ intercepts

Exercise

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