How many solutions will each system of linear equations have? Match the systems with the correct number of solutions.
infinitely many solutions
no solution
one solution
[tex]y=-4 x+11[/tex] and [tex]-6 x+y=11[/tex]
[tex]y=-2 x+5[/tex] and [tex]2 x+y=-7[/tex]
[tex]y=x+6[/tex] and [tex]3 x-3 y=-18[/tex]
Answer (2)
System 1: y = − 4 x + 11 and − 6 x + y = 11 has one solution because the slopes are different.
System 2: y = − 2 x + 5 and 2 x + y = − 7 has no solution because the slopes are the same, but the y-intercepts are different.
System 3: y = x + 6 and 3 x − 3 y = − 18 has infinitely many solutions because the equations are identical.
Therefore, the matching is: one solution, no solution, infinitely many solutions. one solution, no solution, infinitely many solutions
Explanation
Understanding the Problem We have three systems of linear equations and three possible number of solutions: infinitely many, no solution, and one solution. We need to determine the number of solutions for each system and match them correctly.
Analyzing System 1 Let's analyze the first system: y = − 4 x + 11 and − 6 x + y = 11 . We can rewrite the second equation as y = 6 x + 11 . Now we have two equations: y = − 4 x + 11 and y = 6 x + 11 . Since the slopes are different ( − 4 and 6 ), the lines intersect at one point. Therefore, this system has one solution.
Analyzing System 2 Now let's analyze the second system: y = − 2 x + 5 and 2 x + y = − 7 . We can rewrite the second equation as y = − 2 x − 7 . Now we have two equations: y = − 2 x + 5 and y = − 2 x − 7 . Since the slopes are the same ( − 2 ) but the y-intercepts are different ( 5 and − 7 ), the lines are parallel and do not intersect. Therefore, this system has no solution.
Analyzing System 3 Finally, let's analyze the third system: y = x + 6 and 3 x − 3 y = − 18 . We can rewrite the second equation as − 3 y = − 3 x − 18 , and then divide by − 3 to get y = x + 6 . Now we have two equations: y = x + 6 and y = x + 6 . Since the equations are identical, the lines are the same, and there are infinitely many solutions.
Final Answer Matching the systems with the number of solutions:
System 1: y = − 4 x + 11 and − 6 x + y = 11 has one solution.
System 2: y = − 2 x + 5 and 2 x + y = − 7 has no solution.
System 3: y = x + 6 and 3 x − 3 y = − 18 has infinitely many solutions.
Examples
Understanding the number of solutions in a system of linear equations is crucial in various real-world applications. For example, in economics, it can help determine if there is a unique equilibrium point in a supply and demand model. In engineering, it can be used to analyze the stability of a system. In computer graphics, it can be used to determine the intersection of lines and planes. Knowing whether a system has one, none, or infinitely many solutions allows us to make informed decisions and predictions in these diverse fields.
System 1 has one solution, System 2 has no solution, and System 3 has infinitely many solutions. Each system was analyzed by comparing slopes and intercepts to determine how many solutions exist. The relationships between the equations reveal whether they intersect, are parallel, or are identical.
;
