Solve the following system of linear equations by addition. Indicate whether the given system of linear equations has one solution, has no solution, or has an infinite number of solutions. If the system has one solution, find the solution.
{y=−4x+21 8x+2y=24 Selecting an option will enable input for any required text boxes. If the selected option does not have any associated text boxes, then no further input is required.
To solve the system of equations by addition, we need to eliminate one variable by manipulating the equations.
We can start by multiplying the first equation by 2 to make the coefficients of y in both equations the same:
2(y) = 2(-4x + 21)
Simplifying this equation, we get:
2y = -8x + 42
Now, the two equations are:
2y = -8x + 42 (Equation 1)
8x + 2y = 24 (Equation 2)
We can now add Equation 1 and Equation 2 to eliminate y:
(2y) + (2y) = (-8x + 42) + (8x + 2y)
Simplifying the equation, we get:
4y = 42
Dividing both sides of the equation by 4, we find the value of y:
y = 42/4
y = 10.5
Now, we substitute the value of y back into one of the original equations to find the value of x. Let's use Equation 1:
y = -4x + 21
Substituting y = 10.5, we get:
10.5 = -4x + 21
Rearranging the equation, we can solve for x:
4x = 21 - 10.5
4x = 10.5
x = 10.5/4
x = 2.625
Therefore, the solution to the given system of equations is x = 2.625 and y = 10.5.
The system of equations has one solution.