Check all that apply. Select the statement(s) that correctly describes the solution to this system of equations:
8x−2y=−4
4x−y=−2
(2 points)
Responses
Solve this system by elimination since it is already in standard form and lined up nicely.
Solve this system by elimination since it is already in standard form and lined up nicely.
There is exactly one solution to this system of linear equations and it is (2, 0).
There are infinite solutions to this system of linear equations.
These lines are parallel, so there is no solution to this system of linear equations..
There is exactly one solution to this system of linear equations and it is (0, -2).
Solve this system by substitution since one of the variables is defined by the other without having to do any math.
There is exactly one solution to this system of linear equations and it is (2, 0).
To determine the solution to the given system of equations, we can use the method of elimination by adding or subtracting the equations to eliminate one of the variables. Let's go through the options one by one:
1) "Solve this system by elimination since it is already in standard form and lined up nicely." - This statement is not relevant to determining the solution and can be disregarded.
2) "There is exactly one solution to this system of linear equations and it is (2, 0)." - To check if this is the correct solution, we substitute the values of x=2 and y=0 into both equations to see if they satisfy the system. Plugging these values into the first equation: 8(2) - 2(0) = 16 - 0 = 16 ≠ -4. Hence, this is not the correct solution.
3) "There are infinite solutions to this system of linear equations." - To check if this is the correct solution, we can compare the slopes of the two equations. Looking at the coefficients of x in both equations, the ratios are 8/4 = 2. The slopes are not equal, so the lines are not parallel and do not have infinite solutions. This statement is not correct.
4) "These lines are parallel, so there is no solution to this system of linear equations." - As explained in option 3, the lines are not parallel, so this statement is not correct.
5) "There is exactly one solution to this system of linear equations and it is (0, -2)." - To check if this is the correct solution, we substitute the values of x=0 and y=-2 into both equations. Plugging these values into the first equation: 8(0) - 2(-2) = 0 + 4 = 4 ≠ -4. Therefore, this is not the correct solution.
6) "Solve this system by substitution since one of the variables is defined by the other without having to do any math." - This method is not applicable in this case since there is no explicit equation relating one variable to the other.
Based on the above analysis, none of the given statements correctly describe the solution to the system of equations. The solution to the system of equations can be found by eliminating one variable. Adding twice the second equation to the first equation, we get 8x - 2y = -4 + 2(-2) = -4 - 4 = -8. Simplifying this gives us 8x - 2y = -8. This equation represents the same line as the first equation in the original system. Hence, the system is dependent, meaning infinitely many solutions exist.
To solve this system of equations, we can use the method of elimination or substitution. Let's go through the options one by one to determine which one(s) correctly describe the solution:
1. Solve this system by elimination since it is already in standard form and lined up nicely.
This statement is incorrect as the system is not in standard form. It needs to have the x and y terms aligned with each other.
2. There is exactly one solution to this system of linear equations, and it is (2, 0).
To determine if this statement is correct, we need to solve the system of equations. Let's use the elimination method to solve:
Multiply the second equation by 2 to eliminate the y term:
8x - 2y = -4
8x - 2y = -4
The two equations are the same, so they represent the same line. Therefore, there are infinite solutions to this system, not just one.
3. There are infinite solutions to this system of linear equations.
This statement is correct based on the explanation above.
4. These lines are parallel, so there is no solution to this system of linear equations.
This statement is incorrect. The lines are not parallel since they represent the SAME line. Therefore, there are infinite solutions.
5. There is exactly one solution to this system of linear equations, and it is (0, -2).
This statement is incorrect. The correct statement would be that there are infinite solutions, not just one.
6. Solve this system by substitution since one of the variables is defined by the other without having to do any math.
This statement is incorrect. While it is true that the second equation can be rewritten as y = 4x + 2, substitution is not necessary to solve the system.
In summary, the correct statement is:
- There are infinite solutions to this system of linear equations.
are you suree
I apologize for the confusion. Let's reevaluate the options and determine the correct statement:
1) "Solve this system by elimination since it is already in standard form and lined up nicely." - This statement is incorrect as the system is not in standard form. It needs to have the x and y terms aligned with each other.
2) "There is exactly one solution to this system of linear equations and it is (2, 0)." - We previously determined that this statement is incorrect because the values of x = 2 and y = 0 do not satisfy both equations.
3) "There are infinite solutions to this system of linear equations." - This statement is correct since both equations represent the same line. When graphed, the lines would overlap completely, indicating that any point on the line would satisfy both equations.
4) "These lines are parallel, so there is no solution to this system of linear equations." - This statement is incorrect. The lines are not parallel since they represent the same line. Hence, there are infinite solutions.
5) "There is exactly one solution to this system of linear equations and it is (0, -2)." - We previously determined that this statement is incorrect because the values of x = 0 and y = -2 do not satisfy both equations.
6) "Solve this system by substitution since one of the variables is defined by the other without having to do any math." - This statement is incorrect. While it is true that the second equation can be rewritten as y = 4x + 2, substitution is not necessary to solve the system.
Based on the reevaluation, the correct statement is:
- There are infinite solutions to this system of linear equations.
is this correct??
I apologize for any confusion. Let's review the options and determine the correct statement:
1) "Solve this system by elimination since it is already in standard form and lined up nicely." - This statement is incorrect as the system is not in standard form. It needs to have the x and y terms aligned with each other.
2) "There is exactly one solution to this system of linear equations and it is (2, 0)." - We previously determined that this statement is not correct because the values of x = 2 and y = 0 do not satisfy both equations.
3) "There are infinite solutions to this system of linear equations." - This statement is correct because when we manipulate the equations, we find that they represent the same line. Therefore, there are infinite solutions.
4) "These lines are parallel, so there is no solution to this system of linear equations." - This statement is not correct. The lines are not parallel since they represent the same line.
5) "There is exactly one solution to this system of linear equations and it is (0, -2)." - We previously determined that this statement is not correct because the values of x = 0 and y = -2 do not satisfy both equations.
6) "Solve this system by substitution since one of the variables is defined by the other without having to do any math." - This statement is not necessary in this case as the equations can be solved using elimination.
Based on the reevaluation, the correct statement is:
- There are infinite solutions to this system of linear equations.
Jim bought some potted plants:
Petunias cost $11 per pot and Begonias cost $15 per pot.
Jim bought 19 potted plants in all and spent a total of $225 on the plants.
Which would be a correct statement?
(1 point)
Responses
Jim bought 15 begonias.
Jim bought 15 begonias.
Jim bought 15 petunias.
Jim bought 15 petunias.
Jim bought 4 petunias.
Jim bought 4 petunias.
Jim bought 18 begonias.
Jim bought 18 begonias.
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To determine the correct statement, let's set up a system of equations based on the information given:
Let x be the number of Petunias purchased.
Let y be the number of Begonias purchased.
From the problem, we can write two equations:
Equation 1: x + y = 19 (the total number of potted plants purchased)
Equation 2: 11x + 15y = 225 (the total cost of the potted plants)
To find the correct statement, we need to solve this system of equations. Please note that there is an error in the given options as there are duplicate answers. The correct options should be:
1) Jim bought 15 begonias.
2) Jim bought 4 petunias.
Let's solve the system to see which options are correct:
Using the substitution method, we can solve Equation 1 for x:
x = 19 - y
Substituting this value of x into Equation 2, we get:
11(19 - y) + 15y = 225
209 - 11y + 15y = 225
209 + 4y = 225
4y = 16
y = 4
Plugging the value of y = 4 back into Equation 1:
x + 4 = 19
x = 15
So, the correct statement is:
- Jim bought 15 petunias.
And the incorrect statement is:
- Jim bought 15 begonias and 4 petunias.
Therefore, the correct statement is:
- Jim bought 15 petunias.