Consider the two equations
4x+22=8x+10
5x+5=5x+4
you put a square around, which form a system with no solution. Solve the equations.
Let's solve the first equation:
4x + 22 = 8x + 10
First, we can simplify the equation by subtracting 4x from both sides:
22 = 4x + 10
Then, we can simplify further by subtracting 10 from both sides:
12 = 4x
Finally, we can solve for x by dividing both sides by 4:
x = 3
Now let's solve the second equation:
5x + 5 = 5x + 4
We can simplify this equation by subtracting 5x from both sides:
5 = 4
Since 5 does not equal 4, we have a contradiction. This means that the system of equations has no solution.
what do you notice about these equations
Upon further examination, I notice that the two equations are inconsistent.
In the first equation, we found that x = 3. However, in the second equation, we found that 5 = 4, which is not true.
This inconsistency implies that the system of equations cannot be satisfied simultaneously. Hence, the system has no solution.
what does it mean to have no solutions?
Having no solutions in a system of equations means that there is no shared point or set of values that satisfy both equations simultaneously. In other words, there is no value or set of values for the variables in the equations that make both equations true at the same time.
Geometrically, if the equations represent lines on a graph, having no solutions means that the lines do not intersect. They may be parallel or coincident (overlapping), but in this case, they are parallel and will never intersect each other.
Algebraically, having no solutions means that the equations are inconsistent, and there is no common solution that satisfies both equations.
To solve the first equation, we will rearrange the terms to isolate the variable x:
4x + 22 = 8x + 10
Subtract 4x from both sides:
22 = 4x + 10
Subtract 10 from both sides:
12 = 4x
Now, divide both sides by 4:
12/4 = x
x = 3
So, the solution to the first equation is x = 3.
Now, let's solve the second equation:
5x + 5 = 5x + 4
Subtract 5x from both sides:
5 = 4
Here, we end up with a contradictory statement, which means that there is no solution to this equation.
Therefore, the system of equations has no solution.
To solve the equations, let's start with the first equation: 4x + 22 = 8x + 10.
First, we can simplify the equation by moving the constants to one side and the variables to the other side. Subtracting 4x from both sides, we have:
22 = 4x + 10.
Next, we can further simplify the equation by subtracting 10 from both sides:
22 - 10 = 4x.
12 = 4x.
Now, to solve for x, we divide both sides by 4:
12 / 4 = x.
3 = x.
So, the solution to the first equation is x = 3.
Let's move on to the second equation: 5x + 5 = 5x + 4.
In this equation, notice that the variable "x" appears on both sides of the equation. Since we cannot solve for x, we can determine that this equation has no solution. This is because if we subtract 5x from both sides, we are left with 5 = 4, which is not true.
Therefore, the system of equations has no solution.