Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.
6, x, plus, y, equals, minus, 7
6x+y=
−7
minus, 24, x, minus, 7, y, equals, 25
−24x−7y=
25
To determine the number of solutions for the given system of equations, we can solve the system by using one of the methods such as substitution or elimination.
Given system of equations:
6x + y = -7
-24x - 7y = 25
Let's use the elimination method to solve the system:
First, let's multiply the first equation by 7 to eliminate the y term:
(7)(6x + y) = (7)(-7)
42x + 7y = -49
Now, let's add this new equation to the second equation:
(-24x - 7y) + (42x + 7y) = 25 + (-49)
18x = -24
Now, we can solve for x:
x = -24/18
x = -4/3
Substitute this value of x into the first equation to solve for y:
6(-4/3) + y = -7
-8 + y = -7
y = -7 + 8
y = 1
So, the solution to the system of equations is x = -4/3 and y = 1. Therefore, there is exactly one solution for the given system of equations.
Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.
minus, 6, x, plus, 5, y, equals, minus, 9
−6x+5y=
−9
minus, 11, x, plus, 10, y, equals, minus, 17
−11x+10y=
−17
To determine the number of solutions for the given system of equations, we can solve the system by using one of the methods such as substitution or elimination.
Given system of equations:
-6x + 5y = -9
-11x + 10y = -17
Let's use the elimination method to solve the system:
First, let's multiply the first equation by 2 and the second equation by 3 to eliminate the x terms:
(2)(-6x + 5y) = (2)(-9)
-12x + 10y = -18
(3)(-11x + 10y) = (3)(-17)
-33x + 30y = -51
Now, let's add these new equations:
(-12x + 10y) + (-33x + 30y) = -18 + (-51)
-45x + 40y = -69
Now, let's divide every term by 5 to simplify the equation:
(-45x + 40y)/5 = -69/5
-9x + 8y = -13.8
Since -13.8 is not an integer, it is clear that there is no solution for this equation.
Therefore, the given system of equations has no solution.
To determine the number of solutions for the given system of equations, we can use the method of solving a system of linear equations.
1. Start by writing the system of equations:
6x + y = -7 (Equation 1)
-24x - 7y = 25 (Equation 2)
2. Now, we have two options:
a. We can solve the system of equations using one of the methods like substitution or elimination to find the values of x and y.
b. We can determine the number of solutions by analyzing the coefficients of x and y.
Let's use the second option and analyze the coefficients:
For a system of linear equations in two variables (x and y), if the ratio of the coefficients of x and y is the same in both equations, the system has either no solution or infinitely many solutions. If the ratio is different, then the system has exactly one solution.
Let's calculate the ratio for x:
For Equation 1: coefficient of x = 6
For Equation 2: coefficient of x = -24
The ratio of the coefficients of x is 6/(-24) = -1/4.
Next, let's calculate the ratio for y:
For Equation 1: coefficient of y = 1
For Equation 2: coefficient of y = -7
The ratio of the coefficients of y is 1/(-7) = -1/7.
Since the coefficients of x and y have different ratios, the system of equations has exactly one solution.
So, the given system of equations has exactly one solution.