Form a system with infinite solutions. Solve the equations. 7x+3=7x+3, 5x+5=5x+4
To form a system with infinite solutions, we need to have two equations that are equivalent or dependent on each other.
In this case, the first equation, 7x + 3 = 7x + 3, is already in this form. This equation is always true, regardless of the value of x. Therefore, any value of x satisfies this equation, resulting in infinite solutions.
As for the second equation, 5x + 5 = 5x + 4, we can begin by simplifying it:
5x + 5 = 5x + 4
By subtracting 5x from both sides, we get:
5 = 4
This equation is never true since 5 is not equal to 4. Therefore, there are no solutions to this equation.
In summary:
The system of equations:
7x + 3 = 7x + 3,
5x + 5 = 5x + 4
Has infinite solutions for the first equation and no solutions for the second equation.
To form a system with infinite solutions, the two equations should be dependent and equivalent to each other. Let's solve the equations step-by-step:
1) 7x + 3 = 7x + 3
In this equation, notice that we have the same term on both sides (7x + 3). This means that both sides of the equation are equal, and we can simplify it to:
7x + 3 = 7x + 3
Now, let's focus on the second equation:
2) 5x + 5 = 5x + 4
Again, we have the same term (5x + 5) on both sides. By subtracting (5x + 5) from both sides, we can simplify it to:
0 = -1
We obtain the equation 0 = -1, which is not true. This means that there is no solution to the system of equations.
Since the equations are not consistent (0 ≠ -1), they cannot be solved simultaneously, and there are no values for x that satisfy both equations. Therefore, this system has no solution.
To determine if a system of equations has infinite solutions, we need to compare the coefficients of the variables and the constants in the equations. If they are the same, then the system will have infinite solutions. Let's analyze the given system:
Equation 1: 7x + 3 = 7x + 3
Equation 2: 5x + 5 = 5x + 4
In both equations, we can observe that the variables have the same coefficient (7 and 5, respectively) on both sides of the equation. Additionally, the constants on both sides of the equations are also the same (3 in equation 1 and 5 in equation 2).
This means that the system has the same equation twice, and therefore, it has an infinite number of solutions. This occurs because both equations are equivalent, meaning they represent the same line on the Cartesian plane.
Thus, any value of x will satisfy this system of equations.