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.
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.