How many solutions does the system of equations have?
3x=-12y+15 and x+4y=5
To determine the number of solutions a system of equations has, we can solve the system and see if we get a unique solution, infinitely many solutions, or no solution at all.
Let's solve the system of equations:
1) 3x = -12y + 15
2) x + 4y = 5
To eliminate the x term, we can multiply equation 2 by 3:
3(x + 4y) = 3(5)
3x + 12y = 15
Now, we have the following system of equations:
1) 3x = -12y + 15
2) 3x + 12y = 15
Subtracting equation 2 from equation 1, we get:
0 = 0
This equation means that 0 is equal to 0, which is always true.
Since the equations are identical and have infinitely many solutions, the system of equations has infinitely many solutions.
To determine the number of solutions for the given system of equations, we can solve them using a method such as substitution or elimination.
Let's solve the system of equations using the substitution method:
Step 1: Solve the first equation for x in terms of y.
3x = -12y + 15
Divide both sides by 3:
x = -4y + 5
Step 2: Substitute the value of x from Step 1 into the second equation:
x + 4y = 5
Replace x with -4y + 5:
(-4y + 5) + 4y = 5
Step 3: Simplify and solve for y:
-4y + 5 + 4y = 5
Combine like terms:
5 = 5
Step 4: Analyze the result:
The equation 5 = 5 is always true. This suggests that the system of equations is dependent or infinite solutions.
Therefore, the system of equations has infinitely many solutions.
To find the number of solutions for the system of equations, we can solve the equations simultaneously. Let's start by rearranging the equations to solve for x in terms of y.
For the first equation, 3x = -12y + 15, divide both sides by 3 to solve for x: x = (-12y + 15) / 3 = -4y + 5.
For the second equation, x + 4y = 5, subtract 4y from both sides to solve for x: x = 5 - 4y.
Now we have two expressions for x in terms of y:
x = -4y + 5
x = 5 - 4y.
Since both expressions for x are identical, we can conclude that the two original equations represent the same line.
Therefore, the system of equations is dependent or consistent, meaning there are infinitely many solutions. Any pair of values (x, y) that satisfies one of the equations will also satisfy the other equation.