True or false.
1. The system (2x - y = 3) and (x + 3y = 5) is consistent and independent.
2. There are infinitely many solutions to the system (2x - y = 3 and y = 2x + 5).
#1 TRUE
#2 FALSE
How did you come up with the answers true and false, respectively? I want to know so that I can understand it better anonymous. Thanks.
To determine the truth value of the statements, we need to analyze the given systems of equations.
1. The system (2x - y = 3) and (x + 3y = 5) is consistent and independent.
To determine if the system is consistent, we need to check if there are any solutions. To do that, we can solve the system of equations using any method like substitution, elimination, or graphing.
Let's solve the system using the method of substitution:
Start with the first equation: 2x - y = 3
Rearrange it to solve for y: y = 2x - 3
Substitute this expression for y in the second equation:
x + 3(2x - 3) = 5
Simplify the equation: x + 6x - 9 = 5
Combine like terms: 7x - 9 = 5
Add 9 to both sides: 7x = 14
Divide both sides by 7: x = 2
Now that we have found the value of x, we can substitute it back into either of the original equations to find the value of y:
Using the first equation: 2(2) - y = 3
4 - y = 3
Subtract 4 from both sides: -y = -1
Multiply by -1: y = 1
So, the solution to the system is x = 2 and y = 1. Since there is a unique solution, the system is consistent.
To determine if the system is independent, we need to check if the two equations are not multiples of each other. Looking at the two equations, (2x - y = 3) and (x + 3y = 5), we can see that they involve different variables and coefficients. Therefore, the system is independent.
Hence, the statement "The system (2x - y = 3) and (x + 3y = 5) is consistent and independent" is true.
2. There are infinitely many solutions to the system (2x - y = 3) and (y = 2x + 5).
To determine if there are infinitely many solutions, we need to check if the two equations are equivalent. In other words, one equation can be obtained by manipulating the other equation algebraically.
Comparing the two equations, (2x - y = 3) and (y = 2x + 5), we can observe that both equations have different forms and different coefficients.
Since the equations are not equivalent, the system does not have infinitely many solutions.
Hence, the statement "There are infinitely many solutions to the system (2x - y = 3) and (y = 2x + 5)" is false.