How many solutions does the system have?
12x-15y=18
4x-5y=6
To determine the number of solutions, we can rearrange the second equation to solve for x:
4x - 5y = 6
4x = 5y + 6
x = (5y + 6)/4
Now we can substitute this expression for x into the first equation:
12((5y + 6)/4) - 15y = 18
(60y + 72)/4 - 15y = 18
60y + 72 - 60y = 72
Since the equation simplifies to 72 = 72, this implies that the two equations are equivalent and therefore the system has infinite solutions.
To determine the number of solutions for the given system of equations, we need to find whether the two equations are dependent, independent, or inconsistent. This can be done by examining the slopes of the two lines.
First, let's rewrite both equations in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.
12x - 15y = 18
-15y = -12x + 18
y = (12/15)x - (18/15)
y = (4/5)x - (6/5) ... (Equation 1)
4x - 5y = 6
-5y = -4x + 6
y = (4/5)x - (6/5) ... (Equation 2)
Comparing the two equations, we see that they have the same slope of 4/5. This indicates that the lines are parallel and will never intersect. Therefore, the system of equations is inconsistent, meaning there are no solutions.
To determine the number of solutions that the system of equations has, we need to evaluate whether the two lines intersect, are parallel, or coincident.
We can solve this system of equations using the method of substitution or elimination. Let's use the method of substitution:
Step 1: Solve one equation for one variable in terms of the other variable.
Let's solve the second equation for x:
4x - 5y = 6
4x = 5y + 6
x = (5/4)y + 6/4
x = (5/4)y + 3/2
Step 2: Substitute the expression found in Step 1 for the corresponding variable in the other equation.
Replace x in the first equation with (5/4)y + 3/2:
12x - 15y = 18
12((5/4)y + 3/2) - 15y = 18
(15/4)y + 9 - 15y = 18
(15/4)y - (60/4)y = 18 - 9
-(45/4)y = 9
Step 3: Solve for y.
Multiply both sides of the equation by -4/45 to isolate y:
y = (9)(-4/45)
y = -36/45
y = -4/5
Step 4: Substitute the value of y back into either of the original equations to solve for x.
Let's use the first equation:
12x - 15y = 18
12x - 15(-4/5) = 18
12x + 60/5 = 18
12x + 12 = 18
12x = 18 - 12
12x = 6
x = 6/12
x = 1/2
So, the solution to the system of equations is x = 1/2 and y = -4/5.
Since the system has a unique solution, namely (1/2, -4/5), it can be concluded that the system has exactly one solution.
How many solutions does the system have?
12x-15y=18
4x-5y=6