Determine whether each system of linear equations has a)one and only one solution, b)infinitely many solutions, or c) no solution
2x-4y=5
3x +2y=6
the slope of the first line is 1/2
the slope of the 2nd line is -3/2
How do two lines with different slopes meet ?
To determine whether the system of linear equations has one solution, infinitely many solutions, or no solution, we can use the method of elimination or substitution.
Let's use the method of elimination to solve the system of equations:
Step 1: Multiply the first equation by 3 and the second equation by 2 to make the coefficients of x in both equations the same:
3(2x-4y) = 3(5)
2(3x + 2y) = 2(6)
Simplifying these equations, we get:
6x - 12y = 15
6x + 4y = 12
Step 2: Subtract the first equation from the second equation to eliminate x:
(6x + 4y) - (6x - 12y) = 12 - 15
Simplifying this equation, we get:
16y = -3
Step 3: Solve for y:
y = -3/16
Step 4: Substitute the value of y back into one of the original equations and solve for x:
2x - 4(-3/16) = 5
Simplifying this equation, we get:
2x + 3/2 = 5
Step 5: Solve for x:
2x = 5 - 3/2
2x = 10/2 - 3/2
2x = 7/2
x = 7/4
Therefore, the system of linear equations has one and only one solution, with x = 7/4 and y = -3/16.