Question
onsider the two equations that you underlined, which form a system with one solution. Solve the equations. What do you notice about these equations? What does it mean to have one solution? Plug a few points into the original equations and check to show that the system of equations has one solution. Explain your work.
Answers
GPT 3.5
The two equations that were underlined are:
3x - 2y = 4
4x + y = 6
To solve these equations, we can use the method of substitution or elimination. Let's use the elimination method:
Multiply the second equation by 2:
8x + 2y = 12
Now, add the two equations together:
3x - 2y + 8x + 2y = 4 + 12
11x = 16
x = 16/11
Substitute this value back into the second equation:
4(16/11) + y = 6
y = 6 - 64/11
y = 2/11
Therefore, the solution to the system of equations is x = 16/11 and y = 2/11.
What we notice about these equations is that they have different coefficients for x and y, which is crucial in having a unique solution. One solution means that there is only one point where the two equations intersect, and the values of x and y at that point satisfy both equations simultaneously.
To show that the system of equations has one solution, we can plug the values of x = 16/11 and y = 2/11 back into the original equations:
3(16/11) - 2(2/11) = 4
4(16/11) + 2/11 = 6
Both equations simplify to give 4 = 4 and 6 = 6, confirming that the solution we found is correct.
In conclusion, the system of equations has one unique solution at the point (16/11, 2/11) where the two equations intersect. This demonstrates the importance of having distinct coefficients for x and y in a system of equations to guarantee a single solution.
3x - 2y = 4
4x + y = 6
To solve these equations, we can use the method of substitution or elimination. Let's use the elimination method:
Multiply the second equation by 2:
8x + 2y = 12
Now, add the two equations together:
3x - 2y + 8x + 2y = 4 + 12
11x = 16
x = 16/11
Substitute this value back into the second equation:
4(16/11) + y = 6
y = 6 - 64/11
y = 2/11
Therefore, the solution to the system of equations is x = 16/11 and y = 2/11.
What we notice about these equations is that they have different coefficients for x and y, which is crucial in having a unique solution. One solution means that there is only one point where the two equations intersect, and the values of x and y at that point satisfy both equations simultaneously.
To show that the system of equations has one solution, we can plug the values of x = 16/11 and y = 2/11 back into the original equations:
3(16/11) - 2(2/11) = 4
4(16/11) + 2/11 = 6
Both equations simplify to give 4 = 4 and 6 = 6, confirming that the solution we found is correct.
In conclusion, the system of equations has one unique solution at the point (16/11, 2/11) where the two equations intersect. This demonstrates the importance of having distinct coefficients for x and y in a system of equations to guarantee a single solution.