1.) Write a system of equations that has NO solution.
2.) If you wanted to eliminate the y variables in the system of equations below, what would you multiply the first and second equations by?( Do not solve the equation) 2x-3y=8
7x-4y=11
1.) Oh, dear math enthusiasts, let me entertain you with a system of equations that has no solution:
Equation 1: 2x + 3y = 1
Equation 2: 4x - 6y = -2
Now watch as these equations chase each other in circles, leading to an empty solution. It's like a mathematical version of hide-and-seek but without anyone being found. How elusive!
2.) To eliminate the y variables in the system of equations 2x-3y=8 and 7x-4y=11, you will need an all-star multiplier. Let me unveil it, my dear friends: drumroll, please! Ta-da! 🥁
Multiply the first equation by 4 and the second equation by 3. That should do the trick. Bye-bye, y variables! They'll vanish, leaving us with a delightful equation of only x. So long, letter y!
1.) To write a system of equations that has no solution, you need to create two equations that are parallel and have different y-intercepts. For example:
Equation 1: y = 2x + 3
Equation 2: y = 2x + 7
2.) To eliminate the y variables in the system of equations 2x - 3y = 8 and 7x - 4y = 11, you can multiply the first equation by 4 and the second equation by 3. This will result in the same coefficient for the y term in both equations, allowing you to eliminate the y variables when subtracting one equation from the other. However, remember that you are only asked what you would multiply the equations by, not to solve the equation, so the final result is not necessary.
1.) To create a system of equations that has no solution, you need to set up a situation where the two equations are contradictory or represent parallel lines. Let's consider an example:
Equation 1: 2x + 3y = 10
Equation 2: 4x + 6y = 20
To show that this system has no solution, we can multiply Equation 1 by a constant and add it to Equation 2. Let's choose to multiply Equation 1 by -2:
-2(2x + 3y) = -2(10)
-4x - 6y = -20
Now, we can add this modified Equation 1 to Equation 2:
(-4x - 6y) + (4x + 6y) = -20 + 20
0 = 0
As you can see, the resulting equation is always true (0 = 0). This means that the system of equations is dependent and has infinitely many solutions rather than having no solution.
2.) To eliminate the y-variables in the system of equations given, you would multiply the first equation by a constant that ensures the coefficients of the y-variables in both equations cancel each other out. In this case:
Equation 1: 2x - 3y = 8
Equation 2: 7x - 4y = 11
To eliminate the y-variables, you need to make the coefficients of the y-terms equal but with opposite signs. In other words, you want the coefficients to add up to zero. For this system, multiplying the first equation by 4 and the second equation by 3 achieves this:
4(2x - 3y) = 4(8)
3(7x - 4y) = 3(11)
Now the system becomes:
8x - 12y = 32
21x - 12y = 33
By doing this, the coefficients of the y-terms (-12y) in both equations are equal but have opposite signs. This technique allows you to eliminate the y-variables when solving the system.