The point (-4, 6) is a solution to which system?
Responses
{−5x+y=−33x−8y=24
{
−
5
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+
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=
−
3
3
�
−
8
�
=
24
{x+y=2−x+2y=16
{
�
+
�
=
2
−
�
+
2
�
=
16
{−4x+y=6−5x−y=21
{
−
4
�
+
�
=
6
−
5
�
−
�
=
21
{−5x+y=−2−3x+6y=−12
−5x+y=−2
−3x+6y=−12
}
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Match the system of equations to the number of solutions it has.
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y=5x+7
3y−15x=18
x−2y=6
3x−6y=18
y = 3x + 6
y=−13x−4
y=23x−1
y=23x−2
Infinite Solutions No Solutions One Solution
y=23x−1: One Solution
y=−13x−4: One Solution
x−2y=6: One Solution
y=5x+7: One Solution
y = 3x + 6: One Solution
3x−6y=18: One Solution
3y−15x=18: One Solution
there all one solution?
Yes, all the systems of equations have only one solution.
how many solutions does this have
y=5x+7
3y−15x=18
The system of equations has only one solution.
You can solve the second equation for y and then substitute it into the first equation:
3y − 15x = 18
3y = 15x + 18
y = 5x + 6
Now you have two equations that are equivalent to the original system:
y = 5x + 7
y = 5x + 6
Since the slopes of these equations are the same and the y-intercepts are different, they intersect at exactly one point. Therefore, the system has one unique solution.
how many solutions does this have
x−2y=6
3x−6y=18
These two equations represent the same line as the second equation is just a multiple of the first equation. So, the system of equations has infinitely many solutions.
You can simplify the second equation by dividing each term by 3:
3x - 6y = 18
x - 2y = 6
As you can see, the two equations are equivalent to each other and represent the same line. Any value of x and y that satisfies one equation automatically satisfies the other. Therefore, there are infinitely many solutions.
how many solutions does this have
y = 3x + 6
y=−13x−4
To solve this system, you can set the two equations equal to each other to eliminate y:
3x + 6 = -13x - 4
Simplifying and solving for x, you get:
16x = -10
x = -10/16
x = -5/8
Now you can substitute this value of x into either equation to solve for y:
y = 3(-5/8) + 6
y = -15/8 + 48/8
y = 33/8
So the system of equations has one unique solution: (-5/8, 33/8).
how many solutions does this one have
y=23x−1
y=23x−2
These two equations represent two parallel lines with the same slope of 23. Since they never intersect, there are no solutions to the system of equations.
If you subtract the two equations, you get:
y - y = 23x - 1 - (23x - 2)
0 = 1
This is a contradiction, so there is no solution that satisfies both equations. Therefore, the system has no solution.
Put the steps in order that are used to solve the following systems of equations by substitution.
{−7x −2y = −13x − 2y = 11
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1. Solve one of the equations for one of the variables. In this case, we can solve the first equation for x:
-7x - 2y = -13
-7x = -13 + 2y
x = (13/7) - (2/7)y
2. Substitute the expression for the solved variable into the other equation:
-13x - 2y = 11
-13((13/7) - (2/7)y) - 2y = 11
3. Solve for the remaining variable, which in this case is y:
y = -1
4. Substitute y = -1 back into one of the original equations and solve for x:
-7x - 2(-1) = -13
-7x + 2 = -13
-7x = -15
x = 15/7
Therefore, the solution of the system of equations is (15/7, -1).
Jane has $50 and Ellis has $24. Jane is saving $4 per day and Ellis is saving $6 per day.
What equation represents how much money Jane will have in total after saving for d days?
What equation represents how much money Ellis will have in total after saving for d days?
What equation would be used to find out how many it will take for Jane and Ellis to have the same amount of money saved?
After the amount of time found above, how much will Jane and Ellis each have saved?
The equation that represents how much money Jane will have in total after saving for d days is:
J(d) = 4d + 50
Where J(d) represents the amount of money Jane will have saved after d days of saving.
The equation that represents how much money Ellis will have in total after saving for d days is:
E(d) = 6d + 24
Where E(d) represents the amount of money Ellis will have saved after d days of saving.
To find out how many days it will take for Jane and Ellis to have the same amount of money saved, we can set the two equations equal to each other and solve for d:
4d + 50 = 6d + 24
Simplifying and solving for d, you get:
d = 13
So it will take 13 days for Jane and Ellis to have the same amount of money saved.
After 13 days, Jane will have saved:
J(13) = 4(13) + 50 = $98
After 13 days, Ellis will have saved:
E(13) = 6(13) + 24 = $102
You are running a concession stand at a basketball game. You are selling pizza slices and sodas. Each pizza slice sells for $2.50 and each soda sells for $1.25. At the end of the night you made a total of $696.25. You sold a total of 384 pizza slices and sodas combined. You must report the number of pizza slices sold and the number of sodas sold. What equations did you use to solve this where P = the number of pizza slices sold and S = the number of sodas sold?
Responses
P + S = 696.25 and P + S = 384
P + S = 696.25 and P + S = 384
2.50P + 1.25S = 696.25 and P + S = 384
2.50P + 1.25S = 696.25 and P + S = 384
2.50 + 1.25 = 696.25 and P + S = 384
2.50 + 1.25 = 696.25 and P + S = 384
2.50S + 1.25P = 384 and P + S = 696.25
The equations used to solve this problem are:
2.50P + 1.25S = 696.25 (equation representing the total money earned from pizza and soda sales)
P + S = 384 (equation representing the total number of pizza slices and sodas sold)
Where P represents the number of pizza slices sold and S represents the number of sodas sold.
We have two equations with two unknowns, which we can solve through substitution or elimination to find the values of P and S.
so whats the answer?
To solve this system of equations, we can use elimination by multiplying the second equation by 1.25 and subtracting it from the first equation:
2.50P + 1.25S = 696.25
-(1.25P + 1.25S = 480)
1.25P = 216.25
P = 173
Now we can substitute this value of P into either equation to solve for S:
P + S = 384
173 + S = 384
S = 211
Therefore, the number of pizza slices sold was 173 and the number of sodas sold was 211.
The Woodson and Baker families went to the Burger Shack for dinner. The Woodsons bought 3 burger meals and 4 hot dog meals for $48. The Bakers bought 6 burger meals and 2 hot dog meals for $60. How much does each meal cost? Show your work.
If x represents the cost of burger meals, then y represents Response area.
The system of linear equations would look like Response area.
How much does a burger meal cost? Response area
How much does a hot dog meal cost? Response area
Let x be the cost of a burger meal, and y be the cost of a hot dog meal. Then we can write the following system of equations:
3x + 4y = 48 (equation representing the cost of meals purchased by the Woodsons)
6x + 2y = 60 (equation representing the cost of meals purchased by the Bakers)
To solve this system of equations, we can use elimination by multiplying the first equation by 2 and subtracting it from the second equation:
6x + 2y = 60
-(6x + 8y = 96)
-6y = -36
y = 6
Now we can substitute this value of y into either equation to solve for x:
3x + 4y = 48
3x + 4(6) = 48
3x + 24 = 48
3x = 24
x = 8
Therefore, a burger meal costs $8 and a hot dog meal costs $6.
Select the statement that correctly describes the solution to this system of equations.
8x−2y=−4
4x−y=−2
Responses
There is no solution.
, There is no solution.
There is exactly one solution at (2,0).
, There is exactly one solution at (2,0).
There are infinitely many solutions.
There are infinitely many solutions.
There is exactly one solution at (0,–2).
The statement that correctly describes the solution to this system of equations is:
"There is exactly one solution at (2,0)."
To solve this system of equations, we can use substitution to solve for one variable in terms of the other and then substitute it into the other equation:
4x - y = -2
y = 4x + 2
Substituting this expression for y into the first equation, we get:
8x - 2(4x + 2) = -4
Simplifying and solving for x, we get:
x = 1/2
Substituting this value of x into the equation for y, we get:
y = 4(1/2) + 2 = 4
Therefore, the solution to the system of equations is (1/2, 4).
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Karen bought some potted plants.
Petunias cost $7 per pot.
Begonias cost $8 per pot.
She bought 22 potted plants and spent a total of $163. How many petunias did Karen buy?
Responses
13
13
9
9
11
11
7
Let x be the number of petunias Karen bought and y be the number of begonias she bought. We can create a system of linear equations to represent the given information:
x + y = 22 (equation representing the total number of potted plants)
7x + 8y = 163 (equation representing the total cost of potted plants)
We need to solve for x, which is the number of petunias Karen bought. To do this, we can use elimination by multiplying the first equation by -7 and adding it to the second equation:
-7x - 7y = -154
7x + 8y = 163
y = 9
Substituting this value of y into the first equation, we get:
x + 9 = 22
x = 13
Therefore, Karen bought 13 petunias and 9 begonias.
The solution to a system of two linear equations is x = 3; y = 9.
How could the intersection of the graphs of the two equations be located on a coordinate grid? Response area
What does the solution to a system of two linear equations mean on the graph? Response area
Can you have more than one solution to a Linear system of equations? Response area
Can you have exactly two solutions to a Linear system of equations? Response area
Can you have no solutions to a Linear system of equations?
How could the intersection of the graphs of the two equations be located on a coordinate grid?
The intersection of the graphs of the two equations represents the solution to the system of equations. If the solution is x = 3 and y = 9, then the intersection of the graphs of the two equations would be located at the point (3, 9) on a coordinate grid.
What does the solution to a system of two linear equations mean on the graph?
The solution to a system of two linear equations represents the point of intersection of the two lines on a graph. Geometrically, it is the point where the two lines intersect each other.
Can you have more than one solution to a Linear system of equations?
No, a linear system of equations can have either one unique solution, no solution, or infinitely many solutions.
Can you have exactly two solutions to a Linear system of equations?
No, a linear system of equations can have either one unique solution, no solution, or infinitely many solutions.
Can you have no solutions to a Linear system of equations?
Yes, it is possible for a linear system of equations to have no solution. When the two lines are parallel to each other, they do not intersect and there is no solution to the system of equations.
How could the intersection of the graphs of the two equations be located on a coordinate grid?
The intersection of the graphs of the two equations represents the solution to the system of equations. To locate the point of intersection on a coordinate grid, you need to plot the graphs of the two equations and find the point where they intersect. This point will be the solution of the system of equations and it will have coordinates (x, y) that satisfy both equations simultaneously.
For example, if the two equations are y = 2x + 1 and y = -x + 5, you can graph them on a coordinate grid by plotting a few points on each line and then drawing a line through those points. The point where the two lines intersect will be the solution to the system of equations. You can find the point of intersection either by using algebraic methods like substitution or elimination, or by graphing the equations and visually identifying the point of intersection.
Solve the system: y = −x−1
y=3x−5
Responses
(0, -1)
, (0, -1)
(2, 1)
, (2, 1)
(-1, 1)
, (-1, 1)
(1, -2)
To solve this system of equations, we can equate the two expressions for y and solve for x:
-x - 1 = 3x - 5
Simplifying and solving for x, we get:
x = 2
Now we can substitute this value of x into one of the original equations (such as y = -x - 1) to solve for y:
y = -x - 1
y = -2 - 1
y = -3
Therefore, the solution to the system of equations is (2, -3).
what was the answer of the multiple choice
bot?
The point (-4, 6) is a solution to which system?
Responses
{−5x+y=−33x−8y=24
{−4x+y=6−5x−y=21
{x+y=2−x+2y=16
{−5x+y=−2−3x+6y=−12