How many solutions does the system of equations have? 3x = –12y + 15 and x + 4y = 5 (1 point) Responses one one two two infinitely many infinitely many none
The system of equations can be simplified as follows:
3x = -12y + 15 -> x = -4y + 5
x + 4y = 5
We can substitute x in the second equation with -4y + 5:
(-4y + 5) + 4y = 5
-4y + 5 + 4y = 5
0y + 5 = 5
5 = 5
This is a true statement, which means that the two equations are equivalent. Therefore, the system of equations has infinitely many solutions.
The answer is: infinitely many
To determine how many solutions the system of equations has, we can use the method of solving simultaneous equations.
First, let's solve the system using the method of substitution:
From the second equation, we can isolate x:
x = 5 - 4y
Now, substitute this value of x into the first equation:
3(5 - 4y) = -12y + 15
Simplify:
15 - 12y = -12y + 15
The variable -12y cancels out on both sides, leaving us with:
15 = 15
Since the equation is always true, this implies that the two equations are equivalent. Therefore, the system has infinitely many solutions.
Answer: The system of equations has infinitely many solutions.
To determine the number of solutions for the system of equations, we will use the method of substitution or elimination.
Let's start with substitution:
Step 1: Solve one of the equations for one variable.
x + 4y = 5
=> x = 5 - 4y
Step 2: Substitute the expression for x in the other equation.
3x = -12y + 15
3(5 - 4y) = -12y + 15
15 - 12y = -12y + 15
Step 3: Simplify and solve for y.
15 - 12y = -12y + 15
-12y + 12y = -15 + 15
0 = 0
Since the equation simplifies to 0 = 0, we have an identity. This means that the two equations of the system are equivalent. The system of equations is, therefore, dependent, and has infinitely many solutions.
So, the answer is: The system of equations has infinitely many solutions.