Plug a few points into 2x+3=2x+7,5x+5=5x+4 then check to show that they can have no solution. Explain your work.
If we consider the equation 2x + 3 = 2x + 7, we can plug in any value for x and see that the equation is always true. For example, if we choose x = 1, we would have:
2(1) + 3 = 2(1) + 7
2 + 3 = 2 + 7
5 = 9 (which is false)
No matter what value we choose for x, the equation will always result in a false statement. Therefore, this equation has no solution.
Similarly, if we consider the equation 5x + 5 = 5x + 4 and plug in any value for x, we can see that the equation is never true. For example, if we choose x = 1:
5(1) + 5 = 5(1) + 4
5 + 5 = 5 + 4
10 = 9 (which is false)
No matter what value we choose for x, the equation will always result in a false statement. Therefore, this equation also has no solution.
In both cases, the equations are such that the variable x cancels out on both sides of the equation, resulting in an equation that is always false. This means there is no value of x that will satisfy the equation, hence no solution.
To check whether the two given equations have no solution, we need to plug in a few points and see if they satisfy the equations. Let's start with the first equation, 2x + 3 = 2x + 7.
Let's choose x = 0:
2(0) + 3 = 2(0) + 7
0 + 3 = 0 + 7
3 = 7
As we can see, this equation is not satisfied since 3 is not equal to 7. No matter which other point we choose, we will always end up with an equation that cannot be true.
Now, let's move on to the second equation, 5x + 5 = 5x + 4.
Again, let's choose x = 0:
5(0) + 5 = 5(0) + 4
0 + 5 = 0 + 4
5 = 4
Similarly, this equation is not satisfied since 5 is not equal to 4. Regardless of the point we substitute, we will not be able to find a solution that makes the equation true.
Therefore, both equations have no solution because they lead to contradictions when we plug in different points for x.
To check if a system of linear equations has no solution, we need to determine if the lines represented by the equations are parallel. In other words, we need to verify if the slopes of the lines are equal and if the y-intercepts are different.
Let's start by plugging in a few points into the first equation, which is 2x + 3 = 2x + 7.
If we set x = 0, we get:
2(0) + 3 = 2(0) + 7
3 = 7
This is not true. Now let's try setting x = 1:
2(1) + 3 = 2(1) + 7
5 = 9
Again, this is not true. No matter what value we choose for x, we will never get the equation to be true. Therefore, the first equation has no solution.
Now, let's move on to the second equation, which is 5x + 5 = 5x + 4.
If we set x = 0, we get:
5(0) + 5 = 5(0) + 4
5 = 4
Once again, this is not true. Let's try setting x = 1:
5(1) + 5 = 5(1) + 4
10 = 9
This is also not true. Like the first equation, no matter what value we choose for x, we will never get the equation to be true. Therefore, the second equation has no solution as well.
In conclusion, both equations 2x + 3 = 2x + 7 and 5x + 5 = 5x + 4 have no solution.