Drag the tiles to the correct boxes to complete the pairs. Match each system of linear equations with the correct number of solutions.
One solution
No solution
Infinitely many solutions
[tex]
\begin{aligned}
\begin{array}{r}
3 x-y=4 \\
6 x-2 y=8
\end{array} & \longrightarrow \square\\
\begin{array}{r}
y=-4 x-5 \\
y=-4 x+1
\end{array} & \longrightarrow \square\\
\begin{aligned}
-3 x+y=7 \\
2 x-4 y=-8
\end{aligned} &
\end{aligned}
[/tex]
Answer (1)
System 1 has infinitely many solutions because the second equation is a multiple of the first.
System 2 has no solution because the lines are parallel with different y-intercepts.
System 3 has one solution, found by solving the system of equations using elimination, resulting in x = − 2 and y = 1 .
The number of solutions for each system is Infinitely many, No solution, One solution .
Explanation
Analyzing the Systems of Equations We are given three systems of linear equations and we need to determine the number of solutions for each system.
System 1: Infinitely Many Solutions System 1: 3 x − y = 4 and 6 x − 2 y = 8 . Notice that the second equation is simply twice the first equation. This means the two equations represent the same line. Therefore, there are infinitely many solutions.
System 2: No Solutions System 2: y = − 4 x − 5 and y = − 4 x + 1 . These two equations are in slope-intercept form ( y = m x + b ), where m is the slope and b is the y-intercept. Both equations have the same slope ( m = − 4 ) but different y-intercepts ( b = − 5 and b = 1 ). This means the lines are parallel and will never intersect. Therefore, there are no solutions.
System 3: One Solution System 3: − 3 x + y = 7 and 2 x − 4 y = − 8 . We can solve this system using substitution or elimination. Let's use elimination. Multiply the first equation by 4 to get − 12 x + 4 y = 28 . Now add this to the second equation 2 x − 4 y = − 8 : ( − 12 x + 4 y ) + ( 2 x − 4 y ) = 28 + ( − 8 ) − 10 x = 20 x = − 2 Substitute x = − 2 into the first equation: − 3 ( − 2 ) + y = 7 , which simplifies to 6 + y = 7 , so y = 1 . Since we found a unique solution for x and y , there is one solution.
Conclusion Therefore, the systems of equations have the following number of solutions:
System 1: Infinitely many solutions
System 2: No solution
System 3: One solution
Examples
Understanding the number of solutions for a system of linear equations is crucial in various real-world applications. For instance, in economics, it can help determine if there's a unique equilibrium point in a market (one solution), if the market is unstable (no solution), or if there are multiple possible equilibria (infinitely many solutions). Similarly, in engineering, it can help analyze the stability and behavior of systems with multiple interacting components. By analyzing the equations that model these systems, we can predict their behavior and make informed decisions.
