Look at the system of equations below.

4x minus 5y equals 3 line break 3x plus 5y equals 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.

well the dislikes are throwing me off sooo

Well, well, well, let's talk about these methods, shall we?

Substitution can be quite a laborious method for this system. You see, if you choose to solve by substitution, you'll have to solve one equation for one variable and then substitute that expression into the other equation. It's like a game of hide and seek, but with numbers. And trust me, those numbers are experts at hiding!

Now, onto graphing. Picture this: you've got two equations, and you have to plot them on a graph. You'll need to find points of intersection, which means you'll spend quite some time trying to find those tiny little dots where the lines cross each other. It's like a treasure hunt, but instead of gold, you're hunting for the solutions to the system. And let's face it, if you wanted to hunt for hidden things, you could be searching for your car keys right now!

But elimination, oh elimination, it's like the magician of solution methods. It's fast, it's efficient, and it's most certainly the best method for this system. In elimination, you add or subtract the equations in a way that eliminates one variable, giving you a single, lovely equation with only one variable left. It's like watching a magic trick where the magician makes things disappear, except in this case, it's the variables that vanish!

So, to summarize, substitution is like a game of hide and seek, graphing is like a treasure hunt, but elimination? Elimination is like having a magician solve your equations. And who wouldn't want a magician on their side? Trust me, it's the best method for this system!

The student's argument is correct in stating that elimination is the best method for solving this system of equations. Here's why substitution and graphing are less efficient methods in this particular case:

1. Substitution: Substitution involves solving one equation for one variable and then substituting that expression into the other equation. In this system, substitution may be a bit more complicated because both equations have a coefficient of 5 for the variable y. Substituting one equation into the other would require manipulating the equations to eliminate the y-term, which can be more time-consuming.

2. Graphing: Graphing involves plotting the equations on a coordinate plane and finding the point(s) of intersection. While graphing is a reliable method, it might not be the most efficient option here because the system involves decimal coefficients (4 and 5) and non-integer solutions may not be easily determined from the graph.

On the other hand, elimination allows us to eliminate one of the variables by adding or subtracting the equations in a way that cancels out one of the variables. Since the y-coefficients in the two equations are opposites (5 and -5), adding the equations eliminates the y-term. This simplifies the system into a single equation with only one variable, making it easier to solve for either x or y.

The student's argument is 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. This is actually a valid point.

To further support the student's argument, let me explain why substitution and graphing are less efficient methods for solving this system:

1. Substitution method: In the substitution method, you solve one equation for one variable and then substitute that expression into the other equation. This can involve substituting a complex expression into another equation, which may lead to potential mistakes or errors in the calculations. Additionally, in this system, the coefficients of y in both equations are different, making it more challenging to manipulate the equations to solve for a single variable.

2. Graphing method: Graphing involves plotting the equations on a coordinate plane and finding the point where the two graphs intersect, which represents the solution to the system. However, for this specific system, the equations have non-integer coefficients, making it difficult to accurately graph the equations. Additionally, graphing is a more visual and approximate method, and may not provide an exact solution.

On the other hand, elimination method is efficient for this system because when the y-coefficients in the equations have opposite signs, adding the equations eliminates the variable with opposite coefficients. In this case, if we add the two equations, the y-terms will cancel out, allowing us to easily solve for x. Once we have found the value of x, we can then substitute it back into one of the original equations to solve for y.

Hence, the student's argument that elimination is the best method for this system is valid because it simplifies the equations by eliminating one of the variables without involving complex substitutions or potentially inaccurate graphical representations.

"4x minus 5y equals 3 line break 3x plus 5y equals 13"

How about just typing:
4x - 5y = 3
3x + 5y = 13

Don't the y's have opposite coefficients ?
Wouldn't just adding them "eliminate" them?

Just for fun, you should try the other two methods and compare
the steps and the amount of work.