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 that 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 is correct.
When using the elimination method, the goal is to eliminate one variable by adding or subtracting the equations together. In this case, if we add the two equations, the y-terms will cancel out, leaving us with a single equation in terms of x. Since the student correctly pointed out that the y-coefficients have opposite signs, they can be easily eliminated by adding the equations.
On the other hand, substitution involves solving one equation for one variable and substituting it into the other equation. For this specific system, substitution can quickly become time-consuming and complicated. To apply substitution, we need to solve one of the equations for either x or y and substitute it into the other equation. Both equations have similar coefficients, making it harder to directly isolate a variable and substitute it without further manipulations. This makes the substitution method less efficient for solving this system compared to elimination.
Lastly, graphing can also be a less efficient method for solving this system. Graphing requires the representation of both equations on a graph and identifying the intersection point, which represents a solution to the system. For this particular system, the equations have close coefficients and integers, which means the intersection point might not be apparent from a graph and could require extra precision to accurately identify. This makes graphing a less efficient method for solving this system compared to elimination.
Overall, elimination is the most efficient method for solving this system of equations because it allows for the easy elimination of the y-variable due to the opposite signs of the y-coefficients.