Solve using the elimination method.Show work.If the system has no solution or an infinite number of solutions state this.
7x+8y=51,7x+10y=55
To solve the system of equations using the elimination method, we want to eliminate one variable by manipulating the equations so that the coefficients of one variable are the same (or multiples of each other).
Let's start by multiplying the second equation by -1 to change the sign of all its terms:
7x + 8y = 51 (Equation 1)
-7x -10y = -55 (Equation 2)
Now, let's add the two equations together:
(7x + 8y) + (-7x -10y) = 51 + (-55)
7x - 7x + 8y - 10y = -4
-2y = -4
To isolate y, divide both sides of the equation by -2:
-2y / -2 = -4 / -2
y = 2
Now that we know the value of y, we can substitute it into one of the original equations to solve for x. Let's use Equation 1:
7x + 8y = 51
Substituting y = 2:
7x + 8(2) = 51
7x + 16 = 51
Now, isolate x by subtracting 16 from both sides:
7x + 16 - 16 = 51 - 16
7x = 35
Divide both sides by 7 to find x:
7x / 7 = 35 / 7
x = 5
Thus, the solution to the system of equations is x = 5 and y = 2.
There is one unique solution in this case, so there is no infinite number of solutions, and there is no contradiction that would lead to no solution.