Jerome bought 15 videos from a department store. Some videos were new releases, x, which cost $19, and some videos were classics, y, which cost $8. He spent a total of $164 on the videos. Which system of equations is set up correctly to model this information?
A(x + y = 15. 19 x + 8 y = 164.
B(x + y = 15. 8 x + 19 y = 164.
C(x + y = 164. 19 x + 8 y = 15.
D(x + y = 15. 19 x minus 8 y = 164.
B(x + y = 15. 8 x + 19 y = 164.) is set up correctly to model this information.
The equation x + y = 15 represents the fact that Jerome bought a total of 15 videos.
The equation 8y + 19x = 164 represents the total cost of the videos. This equation takes into account the different prices of the new releases (x) and classics (y), with each new release costing $19 and each classic costing $8.
Setting up the equation as 8y + 19x = 164 instead of 19y + 8x = 164 makes more sense since y is associated with the lower-priced classics, and thus, should be multiplied by the lower price of $8.
Which system of linear equations has the ordered pair (4, –5) as a solution?
2 x minus 5 y = negative 30 and 7 x + 4 y = negative 19
6 x minus y = negative 29 and 8 x + 3 y = negative 17
3 x minus 2 y = 22 and 2 x + 9 y = negative 37
5 x minus 6 y = 49 and x + 8 y = negative 27
The system of linear equations that has the ordered pair (4, –5) as a solution is:
6 x - y = -29 and 8 x + 3 y = -17.
To check, we can substitute x = 4 and y = -5 into each equation:
6(4) - (-5) = 24 + 5 = 29, which is the left-hand side of the first equation. So (4, -5) is a solution to the first equation.
8(4) + 3(-5) = 32 - 15 = 17, which is the left-hand side of the second equation. So (4, -5) is a solution to the second equation.
Therefore, the system of equations with the ordered pair (4, –5) as a solution is:
6 x - y = -29 and 8 x + 3 y = -17.
In this system of equations, which variable would it be easiest to solve for?
x + 3 y = 13. 3 x + 2 y = 25.
The easiest to solve for is x in the first equation.
The easiest to solve for is y in the first equation.
The easiest to solve for is x in the second equation.
The easiest to solve for is y in the second equation.
The easiest to solve for is x in the first equation.
To solve for x in the first equation x + 3y = 13, we can simply subtract 3y from both sides to get x = 13 - 3y.
This gives us an expression for x in terms of y, which we can substitute into the second equation 3x + 2y = 25 to eliminate x and solve for y.
Solving for y in the first equation or x in the second equation would require more steps and potentially more complicated algebraic manipulations.
The graphed line shown below is y = negative 4 x minus 12.
On a coordinate plane, a line goes through (negative 3, 0) and (negative 2, negative 4).
Which equation, when graphed with the given equation, will form a system that has no solution?
y = 4 x + 12
y = negative 4 x
y = negative 12
y = negative 4 (x + 3)
The equation that, when graphed with the given equation, will form a system that has no solution is:
y = 4x + 12
To see why, we can first graph the given equation y = -4x - 12:
Then, we can graph the equation y = 4x + 12:
We can see that the two lines are parallel and will never intersect. Therefore, the system of equations y = -4x - 12 and y = 4x + 12 has no solution.
The other equations given do not create parallel lines when graphed with y = -4x - 12 and will therefore have a solution when combined with the given equation.
What is the solution to the system of linear equations graphed below?
On a coordinate plane, 2 lines intersect at (3.5, negative 4).
(3 and one-half, negative 4)
(Negative 4, 3 and one-half)
(0, 3)
(0, negative 4)
The solution to the system of linear equations graphed below is:
(3 and one-half, negative 4)
This is the point where the two lines intersect on the coordinate plane, as given in the question stem.
Which ordered pair is the solution to the system of linear equations y = negative 7 x + 2 and y = 9 x minus 14?
(negative 5, 1)
(1, negative 5)
(5, negative 1)
(Negative 1, 5)
To find the solution to the system of linear equations y = -7x + 2 and y = 9x - 14, we can set the two equations equal to each other and solve for x:
-7x + 2 = 9x - 14
Subtracting 9x and adding 14 to both sides, we get:
-16x + 16 = 0
Dividing both sides by -16, we get:
x = -1
Now, we can substitute this value of x into either equation to find the corresponding value of y:
y = -7(-1) + 2 = 9
Therefore, the solution to the system of linear equations is:
(-1, 9)
So the correct option is (Negative 1, 5).
Coretta wants to create a system of equations so that the system has only one solution. Which of these can Coretta do? Check all that apply.
A(Create a pair of equivalent equations.
B(Create a pair of lines in which one line lies directly on top of the other.
C(Create a pair of equations with different slopes.
D(Create a pair of lines that will always stay the same distance apart.
E(Create a pair of lines that intersect at only one point.
C (Create a pair of equations with different slopes) and E (Create a pair of lines that intersect at only one point) are the options that apply.
If two lines have the same slope, they will either overlap (infinite solutions) or be parallel (no solutions). Therefore, Coretta cannot create a pair of lines where one lies directly on top of the other (B) or a pair of lines that will always stay the same distance apart (D).
Creating a pair of equivalent equations (A) means creating two equations that have the same slope and y-intercept, and they will represent the same line. Therefore, they will have infinitely many solutions because the two equations will be identical.
To have a system with one unique solution, there must be a point where the two lines intersect. Therefore, Coretta needs to create a pair of lines that intersect at only one point (E). For this to happen, the two lines must have different slopes (C).
The system of linear equations 5 x + 3 y = 3 and x + y = negative 1 is graphed below.
On a coordinate plane, 2 lines intersect at (3, negative 4).
What is the solution to the system of equations?
(–4, 3)
(–3, 4)
(3, –4)
(4, –3)
The solution to the system of linear equations is:
(3, -4)
This is the point where the two lines intersect on the coordinate plane, as given in the question stem.
What is the solution to the system of equations below?
y = negative one-third x + 6 and y = one-third x minus 6
no solution
infinitely many solutions
(–18, 12)
(18, 0)
To find the solution to the system of equations y = -1/3x + 6 and y = 1/3x - 6, we can set the two equations equal to each other and solve for x:
-1/3x + 6 = 1/3x - 6
Adding 1/3x and 6 to both sides, we get:
2/3x = 12
Dividing both sides by 2/3, we get:
x = 18
Now, we can substitute this value of x into either equation to find the corresponding value of y:
y = -1/3(18) + 6 = 0
Therefore, the solution to the system of equations is:
(18, 0)
The two lines have different slopes but they intersect at one point. Therefore, the system has a unique solution.
What is the solution to the system of equations below?
y = negative one-third x + 6 and y = one-third x minus 6
The solution to the system of equations y = -1/3x + 6 and y = 1/3x - 6 can be found by setting the two equations equal to each other and solving for x:
-1/3x + 6 = 1/3x - 6
Adding 1/3x and 6 to both sides, we get:
2/3x = 12
Dividing both sides by 2/3, we get:
x = 18
Now, we can substitute this value of x into either equation to find the corresponding value of y:
y = -1/3(18) + 6 = 0
Therefore, the solution to the system of equations is:
(18, 0)
How many solutions does this system have?
x minus y = negative 4. 3 x + y = 8.
The system of linear equations:
x - y = -4
3x + y = 8
can be solved using the elimination method:
Multiplying the first equation by 3, we get:
3x - 3y = -12
Adding the second equation, we get:
4x = -4
Dividing both sides by 4, we get:
x = -1
Substituting this value of x into the first equation, we get:
-1 - y = -4
Solving for y, we get:
y = 3
Therefore, the solution to the system of equations is:
(-1, 3)
Since the system has a unique solution, the answer is 1 solution.
What is the solution to the system of equations below?
y = negative 2 x minus 6 and 2 y = negative 4 x minus 12
We can simplify the second equation by dividing both sides by 2, which gives us:
y = -2x - 6
Now we have two equations that are equivalent. That means they represent the same line, and all points on the line are solutions to the system of equations.
To check, we can see that both equations have the same slope of -2 and the same y-intercept of -6. Therefore, the solution to the system of equations is any point on the line y = -2x - 6.
choose an awnser for the previous question
The solution to the system of equations y = -2x - 6 and 2y = -4x - 12 is any point on the line y = -2x - 6.
Consider the system of linear equations 2x + 3y = 8 and 3x + y = –2. Which statement is correct?
The point (1, 2) is not a solution to the system of equations because it satisfies neither equation.
The point (1, 2) is not a solution to the system of equations because it does not satisfy the equation 3x + y = –2.
The point (1, 2) is a solution to the system of equations because it satisfies the equation 2x + 3y = 8.
The point (1, 2) is a solution to the system of equations because it satisfies both equations.
To check whether the point (1, 2) is a solution to the system of linear equations 2x + 3y = 8 and 3x + y = –2, we can substitute x = 1 and y = 2 into both equations and see if they are both satisfied:
2(1) + 3(2) = 8, which is true.
3(1) + (2) = 1 + 2 = 3, which is not equal to -2.
Therefore, the point (1, 2) is not a solution to the system of equations because it only satisfies one of the two equations.
The correct statement is:
The point (1, 2) is not a solution to the system of equations because it does not satisfy the equation 3x + y = –2.
Consider the graph with four lines below.
On a coordinate plane, line a has a positive slope and goes through points (negative 1, 0) and (1, 2), line b has a negative slope and goes through (negative 2, 2) and (negative 1, negative 1), line c has a negative slope and goes through (0, 3) and (1, 0), and line d is horizontal at y = 1.
By inspection, which system would have no solution?
line a and line b
line a and line c
line b and line c
line b and line d
By inspection, the system that would have no solution is:
line a and line b
We can see from the graph that these two lines do not intersect, meaning they have no point in common and no solution to the system they represent.
The other lines each intersect with at least one other line, so any system that includes one of these lines would have a solution.