Consider 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 = 8
4x - y = 10
To solve this system of equations, we can use the substitution method. First, solve the second equation for y:
y = 4x - 10
Now substitute this value of y into the first equation:
3x + 2(4x - 10) = 8
3x + 8x - 20 = 8
11x - 20 = 8
11x = 28
x = 28/11
Now plug this value of x back into the second equation to solve for y:
4(28/11) - y = 10
112/11 - y = 10
y = 112/11 - 110/11
y = 2/11
Therefore, the solution to the system of equations is x = 28/11 and y = 2/11.
What we notice about these equations is that the coefficients of y in both equations are different. This is a key characteristic that allows these equations to have one solution.
Having one solution means that there is only one point of intersection between the two lines represented by the equations.
To verify that the system of equations has one solution, we can plug the values of x = 28/11 and y = 2/11 back into the original equations:
3(28/11) + 2(2/11) = 8
4(28/11) - 2/11 = 10
After simplifying the equations, we find that both equations hold true for the given values of x and y, confirming that they are the solution to the system of equations.