Look at the system of equations below.
4x-5y=3
3x+5y=13
A student makes this argument: Elimination is the best method for solving this system because the y-coefficient in the first equation is the opposite of the y-coefficient in the second equation.
Complete the student’s argument by explaining why substitution and graphing are less efficient methods than elimination for this system.
The student's argument is correct in stating that elimination is the best method for solving this system because the y-coefficients in the two equations are opposites of each other, which means that when we add the two equations together, the y-term cancels out.
On the other hand, substitution and graphing are less efficient methods for this system.
Substitution involves solving one equation for one variable and substituting it into the other equation. In this case, both equations have a y-term, which means that if we solve one equation for y, it would lead to a more complicated expression involving the other variable x. This can make the substitution process more tedious and prone to errors.
Graphing involves plotting the equations on a coordinate plane and finding the point(s) of intersection, which represent the solutions. However, in this case, the equations have non-integer coefficients, which can make it difficult to accurately graph the equations by hand. Moreover, graphing may not provide an exact solution if the intersection point(s) do not fall on integer coordinates.
Elimination, in comparison, is more efficient and straightforward. By adding or subtracting the two equations (depending on the signs of the coefficients), we can eliminate one variable and solve for the remaining variable. In this case, the y-term can be easily eliminated since its coefficients are opposites. This simplifies the system into a single equation with only one variable, making it easier to solve for the values of x and y.