How many solutions does the following system have:
6x - 3y = 9
2x - y = 3
Explain how to determine the number of solutions without solving the system. Then apply elimination, and interpret the resulting equation.
could you please explain this to me?
number of solutions:no more than the degree of the equation.
In the above, both x,y are degree one, so there is at most one real solution. THere maybe no solutions.
To solve, multiply the lower equation by 3, and you see that they are the same equation, which then has no solution, as the two lines are colinear (the same). It takes graphs that cross each other (intersect) to produce a solution.
To determine the number of solutions without solving the system, we can examine the coefficients of the variables in each equation.
In the given system:
6x - 3y = 9 (equation 1)
2x - y = 3 (equation 2)
To compare the equations, we'll multiply equation 2 by 3 so that the coefficients of y will cancel out when we eliminate the variable y.
Multiplying equation 2 by 3:
3 * (2x - y) = 3 * 3
6x - 3y = 9 (equation 3)
Now, we compare equation 1 and equation 3:
6x - 3y = 9 (equation 1)
6x - 3y = 9 (equation 3)
From this comparison, we can see that equation 1 and equation 3 are identical.
Since equation 1 and equation 3 are the same, it implies that the two original equations (equation 1 and equation 2) represent the same line when graphed on the coordinate plane.
So, the system of equations has infinitely many solutions, since the two equations represent the same line, and any point on this line is a solution to the system.
Interpreting the resulting equation after elimination:
The resulting equation after elimination (equation 3) confirms that the two original equations represent the same line. This means that all the points on that line satisfy both equations, making it an infinite solution system.