Consider the two equations

The first equation, 7x + 3 = 7x + 3, is an identity. This means that the equation is true for all values of x. There are infinitely many solutions to this equation.

The second equation, 3(4x + 8) = 4(3x + 6), can be simplified:

12x + 24 = 12x + 24

When we subtract 12x from both sides and subtract 24 from both sides, we get:

0 = 0

This equation is also an identity, meaning it is true for all values of x. Therefore, there are infinitely many solutions to this equation as well.

In summary, both of the given equations form a system with infinitely many solutions.

what do you notice about these questions?

I notice that both equations are identity equations, which means that they are true for all values of x. This is why there are infinitely many solutions for both equations.

what does it mean to have infinite solutions?

Having infinitely many solutions means that there are an unlimited number of values of x that satisfy the equation. In other words, if you were to plug in any value for x, the equation would still be true. It indicates that the equations are not sufficient to determine a unique value for x. Instead, the solutions form a continuous range or a set of values.

To solve the system of equations, let's focus on the first equation:

7x + 3 = 7x + 3

By subtracting 7x from both sides, we get:

3 = 3

This equation is true.

Now let's consider the second equation:

3(4x + 8) = 4(3x + 6)

Using the distributive property, we can simplify:

12x + 24 = 12x + 24

By subtracting 12x from both sides, we get:

24 = 24

This equation is also true.

Since both equations simplifies to a true statement, it means that the system of equations has infinite solutions. In other words, any value of x will satisfy both equations.

To identify if the system of equations has infinite solutions, we need to check if the two equations are equivalent or result in the same equation.

First, let's solve the first equation:

7x + 3 = 7x + 3

Subtracting 7x from both sides, we get:

3 = 3

Since 3 is equal to 3, this implies that the first equation is true for all values of x. Thus, this equation has infinite solutions.

Now, let's solve the second equation:

3(4x + 8) = 4(3x + 6)

Expanding both sides of the equation, we have:

12x + 24 = 12x + 24

As we can see, the equation simplifies to 12x + 24 = 12x + 24, which means it is always true. Therefore, the second equation also has infinite solutions.

Hence, the system of equations, 7x + 3 = 7x + 3 and 3(4x + 8) = 4(3x + 6), has infinitely many solutions.