x Suppose the system of linear equations has no solution. Which is NOT a possible
value of the constant for the second equation?
6x + 9y = 15
2x + 3y =
x
X
X
X
———
A 5
B 10
C
15
D 20
To determine which value is not possible for the constant in the second equation, we need to understand the conditions under which a system of linear equations has no solution.
A system of linear equations has no solution when the two lines represented by the equations are parallel and never intersect. In this case, the slopes of the lines must be equal, but the y-intercepts must be different.
Let's examine the given system of linear equations:
6x + 9y = 15 (Equation 1)
2x + 3y = x (Equation 2)
To find the slope-intercept form of each equation, we will isolate y on one side:
Equation 1: 9y = -6x + 15 -> y = (-2/3)x + 5/3
Equation 2: 3y = -x -> y = (-1/3)x
Comparing the two equations, we see that the slopes are different (-(2/3) and -(1/3)). Therefore, the second equation can take any constant value, and all of the options provided (5, 10, 15, and 20) are possible.
So, the answer is None of the above.
The possible values for the constant in the second equation can be found by solving the system of equations.
Subtracting 2 times the first equation from the second equation, we have:
(2x + 3y) - 2(6x + 9y) = x - 2(15)
2x + 3y - 12x - 18y = x - 30
-10x - 15y = x - 30
-11x - 15y = -30
In order for the system of equations to have no solution, the lines represented by the equations must be parallel and never intersect. Thus, the slopes of the lines should be equal. The slope of the second equation is (-11/15).
Since the slope of the first equation is (-6/9) = (-2/3), any constant value that results in a different slope for the second equation is not possible.
Looking at the answer choices:
A) 5 -> (-11/15) ≠ (-2/3)
B) 10 -> (-11/15) ≠ (-2/3)
C) 15 -> (-11/15) ≠ (-2/3)
D) 20 -> (-11/15) ≠ (-2/3)
Therefore, none of the given answer choices (A, B, C, or D) are possible values for the constant in the second equation.
To determine the constant that is NOT a possible value for the second equation in a system of linear equations with no solution, we can compare the coefficients of the x and y terms in both equations.
In the first equation:
6x + 9y = 15
And in the second equation:
2x + 3y = ? (constant)
If the two equations have no solution, it means that the lines represented by the equations are parallel and will never intersect. In order for the lines to be parallel, the ratio of the coefficients of x in both equations should be the same as the ratio of the coefficients of y.
Let's calculate the ratio of the coefficients of x and y in the first equation:
Coefficient of x: 6
Coefficient of y: 9
Ratio: 6/9 = 2/3
Now let's calculate the ratio of the coefficients of x and y in the second equation:
Coefficient of x: 2
Coefficient of y: 3
Ratio: 2/3
Since the ratio of the coefficients of x and y in the second equation (2/3) is the same as the ratio in the first equation (2/3), this means that any value for the constant in the second equation is possible, and none of the provided values (5, 10, 15, 20) would make the system of equations have no solution.
Therefore, the answer is: None (X), as no value is not possible for the constant in the second equation.