Question
Plug a few points into the original equations and check to show that the system of
equations has one solution. Explain your work.
3(4x+)=4(3x+6)
4x+22=8x+10
equations has one solution. Explain your work.
3(4x+)=4(3x+6)
4x+22=8x+10
Answers
Answered by
GPT 3.5
To check if the system of equations has one solution, we need to plug a few points into the original equations and see if they hold true.
First, let's check the first equation:
3(4x+1) = 4(3x+6)
Let's plug in x = 1:
3(4(1)+1) = 4(3(1)+6)
3(4+1) = 4(3+6)
3(5) = 4(9)
15 = 36
As 15 is not equal to 36, this point does not satisfy the first equation.
Next, let's check the second equation:
4x+22 = 8x+10
Let's plug in x = 2:
4(2)+22 = 8(2)+10
8+22 = 16+10
30 = 26
Again, 30 is not equal to 26, so this point does not satisfy the second equation.
Since neither of the points we tried satisfied both equations, we can conclude that the system of equations does not have one solution.
First, let's check the first equation:
3(4x+1) = 4(3x+6)
Let's plug in x = 1:
3(4(1)+1) = 4(3(1)+6)
3(4+1) = 4(3+6)
3(5) = 4(9)
15 = 36
As 15 is not equal to 36, this point does not satisfy the first equation.
Next, let's check the second equation:
4x+22 = 8x+10
Let's plug in x = 2:
4(2)+22 = 8(2)+10
8+22 = 16+10
30 = 26
Again, 30 is not equal to 26, so this point does not satisfy the second equation.
Since neither of the points we tried satisfied both equations, we can conclude that the system of equations does not have one solution.
Answer
Plug more than one point into the original equations and check to show that the
equations can have more than one solution. Explain your work
7x+3=7x+3
3x+18=5x+8
equations can have more than one solution. Explain your work
7x+3=7x+3
3x+18=5x+8
Answered by
GPT 3.5
To check if the equations can have more than one solution, we need to plug more than one point into the equations and see if they hold true.
For the first equation:
7x+3 = 7x+3
Let's plug in x = 0:
7(0)+3 = 7(0)+3
3 = 3
The equation holds true for any value of x. This means that any value of x will satisfy the equation. Therefore, there are infinitely many solutions for this equation.
Now, let's check the second equation:
3x+18 = 5x+8
Let's plug in x = 2:
3(2)+18 = 5(2)+8
6+18 = 10+8
24 = 18
In this case, the equation does not hold true. This means that the given point does not satisfy the equation.
Therefore, the second equation does not have a solution for x = 2.
Since we found that the first equation has infinitely many solutions and the second equation has no solution for a particular point, we can conclude that the equations can indeed have more than one solution.
For the first equation:
7x+3 = 7x+3
Let's plug in x = 0:
7(0)+3 = 7(0)+3
3 = 3
The equation holds true for any value of x. This means that any value of x will satisfy the equation. Therefore, there are infinitely many solutions for this equation.
Now, let's check the second equation:
3x+18 = 5x+8
Let's plug in x = 2:
3(2)+18 = 5(2)+8
6+18 = 10+8
24 = 18
In this case, the equation does not hold true. This means that the given point does not satisfy the equation.
Therefore, the second equation does not have a solution for x = 2.
Since we found that the first equation has infinitely many solutions and the second equation has no solution for a particular point, we can conclude that the equations can indeed have more than one solution.
Answer
Plug a few points into the original equations and check to show that they can have no
solution. Explain your work.
2x+3=2x+7
5x+5=5x+4
solution. Explain your work.
2x+3=2x+7
5x+5=5x+4
Answered by
GPT 3.5
To check if the equations can have no solution, we need to plug a few points into the equations and see if they hold true.
For the first equation:
2x+3 = 2x+7
Let's solve for x:
2x = 2x+4
0 = 4
We obtained a contradiction, as 0 is not equal to 4. Therefore, there is no value of x that can satisfy this equation. Thus, the first equation has no solution.
Now, let's check the second equation:
5x+5 = 5x+4
Let's solve for x:
5x = 5x-1
0 = -1
Again, we have a contradiction, as 0 is not equal to -1. Hence, there is no value of x that can satisfy this equation either. Therefore, the second equation has no solution.
By checking a few points, we found that neither of the equations has a solution. Thus, we can conclude that both equations can have no solution.
For the first equation:
2x+3 = 2x+7
Let's solve for x:
2x = 2x+4
0 = 4
We obtained a contradiction, as 0 is not equal to 4. Therefore, there is no value of x that can satisfy this equation. Thus, the first equation has no solution.
Now, let's check the second equation:
5x+5 = 5x+4
Let's solve for x:
5x = 5x-1
0 = -1
Again, we have a contradiction, as 0 is not equal to -1. Hence, there is no value of x that can satisfy this equation either. Therefore, the second equation has no solution.
By checking a few points, we found that neither of the equations has a solution. Thus, we can conclude that both equations can have no solution.
Answer
What does it mean to have one solution?
Answered by
GPT 3.5
Having one solution means that there is a specific value for the variable (or variables) in the equation that satisfies the equation and makes it true. In other words, if a system of equations has one solution, it means there is only one point of intersection between the equations.
Graphically, if we plot the equations on a coordinate plane, having one solution means that the graphs of the equations intersect at a single point. Algebraically, having one solution means that when we solve the system of equations, we find a unique value for each variable that satisfies all the equations simultaneously.
Graphically, if we plot the equations on a coordinate plane, having one solution means that the graphs of the equations intersect at a single point. Algebraically, having one solution means that when we solve the system of equations, we find a unique value for each variable that satisfies all the equations simultaneously.
Answer
What do you notice about equations that are in each of these categories? Use
complete sentences to answer the questions that follow.
2x+3=2x+7
7x+3=7x+3
4x+22=8x+10
5x+5=5x+4
3x+18=5x+8
3(4x+8)=4(3x+6)
complete sentences to answer the questions that follow.
2x+3=2x+7
7x+3=7x+3
4x+22=8x+10
5x+5=5x+4
3x+18=5x+8
3(4x+8)=4(3x+6)
Answered by
GPT 3.5
Based on the equations given, we can notice the following:
1. In the equation 2x+3=2x+7, the variables cancel out from both sides of the equation. This results in a false statement, indicating that there is no solution for this equation.
2. In the equation 7x+3=7x+3, both sides are the same. This implies that the equation is always true, regardless of the value of x. Therefore, there are infinitely many solutions for this equation.
3. In the equation 4x+22=8x+10, we can simplify it by subtracting 4x from both sides, which results in 22 = 4x+10. By rearranging the terms, we have 12 = 4x. Dividing both sides by 4 gives us x = 3. This equation has a single solution, x = 3.
4. In the equation 5x+5=5x+4, simplifying it by subtracting 5x from both sides yields 5 = 4. This is a false statement, indicating that there is no solution for this equation.
5. In the equation 3x+18 = 5x+8, we can simplify it by subtracting 3x from both sides, which gives us 18 = 2x+8. By rearranging the terms, we find 10 = 2x. Dividing both sides by 2 yields x = 5. This equation has a single solution, x = 5.
6. In the equation 3(4x+8)=4(3x+6), we can distribute the multiplication on both sides to simplify it to 12x+24 = 12x+24. Both sides of this equation are the same, which means there are infinitely many solutions.
To summarize, we can observe that some equations have no solution, some have infinitely many solutions, and some have a single solution. These categories depend on the relationships between the coefficients and constants in the equations.
1. In the equation 2x+3=2x+7, the variables cancel out from both sides of the equation. This results in a false statement, indicating that there is no solution for this equation.
2. In the equation 7x+3=7x+3, both sides are the same. This implies that the equation is always true, regardless of the value of x. Therefore, there are infinitely many solutions for this equation.
3. In the equation 4x+22=8x+10, we can simplify it by subtracting 4x from both sides, which results in 22 = 4x+10. By rearranging the terms, we have 12 = 4x. Dividing both sides by 4 gives us x = 3. This equation has a single solution, x = 3.
4. In the equation 5x+5=5x+4, simplifying it by subtracting 5x from both sides yields 5 = 4. This is a false statement, indicating that there is no solution for this equation.
5. In the equation 3x+18 = 5x+8, we can simplify it by subtracting 3x from both sides, which gives us 18 = 2x+8. By rearranging the terms, we find 10 = 2x. Dividing both sides by 2 yields x = 5. This equation has a single solution, x = 5.
6. In the equation 3(4x+8)=4(3x+6), we can distribute the multiplication on both sides to simplify it to 12x+24 = 12x+24. Both sides of this equation are the same, which means there are infinitely many solutions.
To summarize, we can observe that some equations have no solution, some have infinitely many solutions, and some have a single solution. These categories depend on the relationships between the coefficients and constants in the equations.