How many solutions does the system of equations have? 4x = -12y + 16 and x + 3y = 4
4 x = -12 y + 16
Divide both sides by 4
x = - 3 y + 4
Add 3 y to both sides
x + 3 y = - 3 y + 4 + 3 y
x + 3 y = 4
This mean 4 x = -12 y + 16 and x + 3 y = 4 is the same equation.
Infinite number of solutions.
That is true of all values of x and y.
Let's solve this system of equations and find out how many solutions it has. But first, let me juggle these numbers to lighten the mood...
Alright, let's get down to business! We'll start by rearranging the first equation to solve for x.
4x = -12y + 16
Dividing both sides by 4, we get:
x = -3y + 4
Now let's substitute this value of x into the second equation and solve for y.
x + 3y = 4
Replacing x with -3y + 4, we get:
-3y + 4 + 3y = 4
Simplifying, we have:
4 = 4
Well, well, well! This equation simplifies to 4 = 4, which is always true. It means that the two equations represent the same line, thus having infinitely many solutions. Kind of like a bottomless pie! So in this case, the system has more solutions than you can count on one hand.
To determine the number of solutions, we can solve the system of equations using any method, such as substitution or elimination. Let's use the method of substitution.
First, we solve the second equation for x:
x + 3y = 4
x = 4 - 3y
Next, we substitute the value of x in the first equation:
4x = -12y + 16
4(4 - 3y) = -12y + 16
16 - 12y = -12y + 16
By simplifying the equation, we find that -12y and 12y would cancel out, leaving us with:
16 = 16
This equation is true, and it means that the two equations are actually the same line. Therefore, the system of equations has infinitely many solutions.
In conclusion, the system of equations has infinitely many solutions.
To determine the number of solutions for the given system of equations, we can use the concept of linear independence. If the system of equations has a unique solution, it is called consistent and independent. If it has infinitely many solutions, it is consistent and dependent. If there are no solutions, it is inconsistent.
Let's solve the system of equations step by step using the method of substitution:
1) Begin with the first equation: 4x = -12y + 16
We can solve this equation for x:
Divide both sides by 4:
x = -3y + 4
2) Substitute the value of x (-3y + 4) into the second equation:
x + 3y = 4
(-3y + 4) + 3y = 4
Simplify:
-3y + 4 + 3y = 4
-3y + 3y + 4 = 4
4 = 4
Since the equation simplifies to 4 = 4, this is a consistent and dependent system of equations. It means that there are infinitely many solutions.
To see why, let's consider the two equations in slope-intercept form:
Equation 1: x = -3y + 4
Equation 2: x + 3y = 4
Equation 1 represents a line with a slope of -3 and a y-intercept of 4.
Equation 2 also represents a line.
Since the lines are not parallel, they intersect at a single point. However, that does not mean that the system has a unique solution. Instead, it means that every point on the line defined by Equation 2 satisfies both equations. Hence, there are infinitely many solutions to this system of equations.
Therefore, the given system of equations has infinitely many solutions.