How many solutions does this system have 12x-15y=18 and 4x-5y=6
To find the number of solutions in this system, we can use the concept of cramer's rule. The system of equations can be represented in matrix form as:
[[12, -15], [4, -5]] * [x, y] = [18, 6]
Using Cramer's rule, the determinant of the coefficient matrix is calculated as follows:
|12 -15|
|4 -5| = (12*-5) - (-15*4) = (-60) - (-60) = 0
Since the determinant is zero, this indicates that the system has infinitely many solutions or no solution.
To determine the number of solutions for a system of equations, we can use the concept of linear dependence.
First, let's rewrite the given system of equations:
Equation 1: 12x - 15y = 18
Equation 2: 4x - 5y = 6
We can simplify Equation 1 by dividing both sides by 3:
(12x - 15y) / 3 = 18 / 3
4x - 5y = 6
Comparing Equation 2 and the simplified Equation 1, we can see that they are the same equation. This implies that these two equations are equivalent.
Whenever two equations in a system are equivalent, it means that they represent the same line on a graph. This indicates that the system has infinitely many solutions.
Thus, the given system of equations has infinitely many solutions.
To determine the number of solutions for this system of equations, we can use the concept of lines and their intersection points.
First, let's express the system of equations in standard form:
12x - 15y = 18 (Equation 1)
4x - 5y = 6 (Equation 2)
To find the number of solutions, we need to examine the relationship between the two lines represented by these equations.
If the lines intersect at a single point, the system has one unique solution.
If the lines are parallel and do not intersect, the system has no solutions.
If the lines overlap and coincide, the system has infinitely many solutions.
To simplify the equations, we can divide both sides of Equation 1 by 3 and Equation 2 by 2:
4x - 5y = 6 (Equation 1 simplified)
2x - 2.5y = 3 (Equation 2 simplified)
Now, let's compare the slopes of the two lines by putting them in slope-intercept form (y = mx + b):
In Equation 1, rearrange to solve for y:
4x - 5y = 6
-5y = -4x + 6
y = (4/5)x - 6/5
The slope of this line is (4/5).
In Equation 2, rearrange to solve for y:
2x - 2.5y = 3
-2.5y = -2x + 3
y = (2/5)x - 6/5
The slope of this line is (2/5).
Since the slopes of the two lines are different [(4/5) ≠ (2/5)], the lines are not parallel.
To determine if the lines intersect or coincide, we can compare their y-intercepts.
In Equation 1:
y = (4/5)x - 6/5
The y-intercept is -6/5.
In Equation 2:
y = (2/5)x - 6/5
The y-intercept is also -6/5.
The y-intercepts are the same, indicating that the lines overlap (i.e., they coincide).
Thus, this system of equations has infinitely many solutions.