7. Writing Solve the system
3 5
2 5
x y
x y
using elimination. Explain how you
can check the solution both algebraically and graphically.
8. Open-Ended Write a system of equations that can be solved using
elimination without multiplication.
9. There are 72 members of the show choir. There are 6 more boys than girls in the
choir.
a. Write the model of a system for the above situation.
b. Do you need to multiply any of the equations by a constant before solving
by elimination? Explain.
10. Writing Explain the process you use to determine which variable is the best
variable to eliminate in a system of two equations in two variables.
7. To solve the system using elimination, we can start by multiplying the first equation by -2 and adding it to the second equation:
-2(3x + 5y) + (2x - 5y) = -2y - 6x - 10y + 2x - 5y = -15y - 4x = 0
Simplifying this equation gives us:
-15y - 4x = 0
Now we can add this equation to the first equation:
(3x + 5y) + (-15y - 4x) = 3x - 4x + 5y - 15y = -9x - 10y = 0
Simplifying this equation gives us:
-9x - 10y = 0
Now we have the two equations:
-15y - 4x = 0
-9x - 10y = 0
To check the solution algebraically, we substitute the values we found for x and y back into the original equations and see if both equations are satisfied.
To check the solution graphically, we can plot the two lines representing the equations on a coordinate plane and see if they intersect at the solution point.
8. A system of equations that can be solved using elimination without multiplication:
2x + 3y = 10
3x - 2y = 5
By adding the two equations together, we can eliminate one variable without multiplication:
(2x + 3y) + (3x - 2y) = 2x + 3x + 3y - 2y = 5x + y = 15
9. a. The model of a system for the given situation would be:
x + y = 72 (total number of members)
x = y + 6 (6 more boys than girls)
b. No, we do not need to multiply any of the equations by a constant before solving by elimination because the coefficients of both variables in the second equation are already the same.
10. When determining which variable to eliminate in a system of two equations in two variables, we generally choose the variable that will lead to the simplest elimination. This means looking for variables that have coefficients that are the same or multiples of each other in one equation or can easily be made the same through multiplication or addition/subtraction.