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.
help :,)
The student's argument is correct. The elimination method is the most efficient method for solving this system of equations because the y-coefficients in the two equations are opposites of each other.
Substitution method involves solving one equation for one variable and substituting that expression into the other equation. In this case, substituting the expression for x or y from one equation into the other equation will result in complex expressions. It would be more time-consuming and prone to errors to simplify and solve such expressions.
Graphing method involves plotting the equations on a graph and finding the point of intersection, which represents the solution to the system. Although graphing can provide a visual representation, it is not always accurate and may not provide an exact solution. In addition, it can be difficult to determine the exact coordinates of the intersection point from the graph, especially if the lines are not easily distinguishable or intersect at a fractional point.
On the other hand, the elimination method allows for a direct cancellation of terms. By adding or subtracting the two equations, the terms containing the variable being eliminated get cancelled out, resulting in an equation with one variable which can be easily solved. In this case, adding the two equations eliminates the y variable, and solving the resulting equation gives the value of x. Substituting this value back into either of the original equations helps to find the corresponding value of y.
Hence, elimination is the most efficient method for solving this system of equations as it simplifies the process by eliminating the need for complex substitutions or graphical interpretations.
give me the answer bot
The solution to the system of equations is x = 4 and y = -1.
give me the answer with steps
Sure!
To solve the system of equations using elimination, we can add the two equations together to eliminate the y variable.
Step 1: Add the two equations together:
(4x - 5y) + (3x + 5y) = 3 + 13
7x = 16
Step 2: Solve for x by dividing both sides of the equation by 7:
7x/7 = 16/7
x = 16/7
Step 3: Substitute the value of x back into one of the original equations to solve for y. Let's use the first equation:
4(16/7) - 5y = 3
Step 4: Simplify:
64/7 - 5y = 3
Step 5: Subtract 64/7 from both sides of the equation:
64/7 - 64/7 - 5y = 3 - 64/7
-5y = (21 - 64)/7
-5y = -43/7
Step 6: Divide both sides of the equation by -5:
(-5y)/-5 = (-43/7)/-5
y = 43/35
Thus, the solution to the system of equations is x = 16/7 and y = 43/35.
check your work
My apologies for the previous incorrect response. Let me solve the system of equations correctly:
Given equations:
4x - 5y = 3 --(1)
3x + 5y = 13 --(2)
To solve the system using elimination, we can add the two equations together to eliminate the y variable:
(4x - 5y) + (3x + 5y) = 3 + 13
7x = 16
Divide both sides of the equation by 7 to solve for x:
x = 16/7
Now, substitute the value of x back into either of the original equations to solve for y. Let's use equation (1):
4(16/7) - 5y = 3
Simplify:
64/7 - 5y = 3
Subtract 64/7 from both sides of the equation:
-5y = 3 - 64/7
-5y = 21/7 - 64/7
-5y = (21-64)/7
-5y = -43/7
Divide both sides of the equation by -5 to solve for y:
y = (-43/7) / (-5)
y = 43/35
Hence, the correct solution to the system of equations is x = 16/7 and y = 43/35.