How many solutions does the system have? 12x−15y=18 4x−5y=6 (1 point) Responses no solution no solution two solutions two solutions one solution one solution infinitely many solutions
This system of equations can be solved using the method of substitution or elimination. Let's use the method of substitution.
From the first equation, we can isolate x:
12x - 15y = 18
12x = 18 + 15y
x = 1.5 + 1.25y
Next, substitute the value of x in the second equation:
4(1.5 + 1.25y) - 5y = 6
6 + 5y - 5y = 6
6 = 6
Since the equation 6 = 6 is always true, the system has infinitely many solutions.
To determine the number of solutions for a system of equations, we can use different methods such as graphing, substitution, or elimination.
Let's solve this system of equations using elimination:
Given:
12x - 15y = 18 ...(1)
4x - 5y = 6 ...(2)
To eliminate one variable, we need to multiply both equations by appropriate numbers so that the coefficients of any one variable cancel each other out when added or subtracted.
Let's multiply equation (2) by 3:
3 * (4x - 5y) = 3 * 6
12x - 15y = 18 ...(3) (obtained by multiplying both sides of equation (2) by 3)
Now, we can see that equation (1) and equation (3) are the same:
12x - 15y = 18
12x - 15y = 18
Since the two equations represent the same line, they have infinitely many solutions.
Hence, the answer is: infinitely many solutions.
To determine the number of solutions for a system of equations, we need to find where the two equations intersect on a graph.
Let's solve the system of equations to determine the number of solutions.
Given system of equations:
1) 12x - 15y = 18
2) 4x - 5y = 6
We can solve this system by using the method of elimination or substitution.
Method 1: Elimination
To eliminate a variable, we need to multiply the equations by certain values so that when added or subtracted, one of the variables cancels out.
Multiply equation 2 by 3 to have the same coefficient of x as equation 1:
3 * (4x - 5y) = 3 * 6
12x - 15y = 18
Now we can see that the two equations are actually the same, indicating that they represent the same line on a graph.
Since the two equations represent the same line, they have infinitely many solutions.
Therefore, the answer is infinitely many solutions.