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Question
Solve the system of equations y = 3x - 5 and y = 13
x + 3 using a graphical method.
Responses
A (4, 3)(4, 3)
B (3, 4)(3, 4)
C (0, 3)(0, 3)
D (0, -5)5 of 205 of 20 Items
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Question
Solve the system of equations y = 3x - 5 and y = 13
x + 3 using a graphical method.
Responses
A (4, 3)(4, 3)
B (3, 4)(3, 4)
C (0, 3)(0, 3)
D (0, -5)
To solve the system of equations using a graphical method, we need to plot the graphs of the two equations and determine the point where they intersect, since that point represents the solution to the system.
First, let's plot the graph of y = 3x - 5. This is a linear equation in slope-intercept form, where the slope is 3 and the y-intercept is -5. To graph it, we can plot two points and draw a line through them.
To find the x-intercept, set y = 0:
0 = 3x - 5
3x = 5
x = 5/3 ≈ 1.67
To find the y-intercept, set x = 0:
y = 3(0) - 5
y = -5
So, we have two points: (0,-5) and (5/3,0). Let's plot them:
Next, let's plot the graph of y = 13x + 3. Again, this is a linear equation in slope-intercept form, where the slope is 13 and the y-intercept is 3. To graph it, we can plot two points and draw a line through them.
To find the x-intercept, set y = 0:
0 = 13x + 3
13x = -3
x = -3/13 ≈ -0.23
To find the y-intercept, set x = 0:
y = 13(0) + 3
y = 3
So, we have two points: (0,3) and (-3/13,0). Let's plot them:
Now, we can see that the two lines intersect at the point (4,3). Therefore, the solution to the system of equations is x = 4 and y = 3.
The correct answer is A) (4, 3).
Domain Test
Question Id : 317163
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Question
Rectangle CDEF undergoes a dilation centered at the origin. The result is rectangle C'D'E'F'. Which rule describes the dilation?
Responses
A (x, y) → (2x, 2y)(x, y) → (2x, 2y)
B (x, y) → (4x, 4y)(x, y) → (4x, 4y)
C (x, y) → (−12
x, −12
y)(x, y) → (− 1 2 x, − 1 2 y)
D (x, y) → (12
x, 12
y)
To determine the rule that describes the dilation of rectangle CDEF to C'D'E'F', we need to analyze the changes in the coordinates of the vertices.
A dilation is a transformation that changes the size of a shape while preserving its shape and proportions. In this case, we can see that the dilation of rectangle CDEF results in a rectangle that is larger.
Let's focus on one vertex of the rectangle, such as vertex C. The coordinates of vertex C are (x, y).
Based on the dilation, the x-coordinate is doubled, and the y-coordinate is also doubled. This is because the dilation is centered at the origin, so the origin (0,0) remains the same.
Therefore, the rule that describes the dilation is A) (x, y) → (2x, 2y). This means that the x-coordinate and y-coordinate of each vertex of the original rectangle are multiplied by 2 to determine the coordinates of the corresponding vertex in the dilated rectangle.
The correct answer is A) (x, y) → (2x, 2y).
Porportional relationship
A proportional relationship is a relationship between two variables in which their ratios remain constant. In other words, when one variable increases or decreases, the other variable changes in direct proportion.
For example, consider the relationship between the number of hours worked and the amount earned. If someone earns $10 per hour, the relationship between the number of hours worked (x) and the amount earned (y) can be represented as y = 10x. In this case, the ratio of the amount earned to the number of hours worked remains constant at $10 per hour.
Another example is the relationship between distance and time when traveling at a constant speed. If you are traveling at a speed of 60 miles per hour, the relationship between the distance traveled (x) and the time taken (y) can be represented as y = 60x. Again, the ratio of the distance traveled to the time taken remains constant at 60 miles per hour.
In a proportional relationship, if you were to graph the relationship, the points would lie on a straight line that passes through the origin (0,0). Additionally, the equation representing the relationship would be in the form y = kx, where k is the constant of proportionality.
It's important to note that not all relationships are proportional. In a non-proportional relationship, the ratios between the variables do not remain constant as one variable changes.
Understanding proportional relationships can be helpful in various real-life scenarios, such as calculating rates, scaling objects, determining unit rates, and analyzing linear relationships.
To solve the system of equations y = 3x - 5 and y = 13x + 3 using a graphical method, we need to plot the equations on a graph and find the point of intersection.
First, let's graph the equation y = 3x - 5:
- Plot a point at (0, -5).
- From that point, use the slope of 3 (which is the coefficient of x) to find another point. Move 1 unit to the right and 3 units up from the initial point. Plot this point at (1, -2).
- Connect the two points with a straight line.
Next, let's graph the equation y = 13x + 3:
- Plot a point at (0, 3).
- From that point, use the slope of 13 to find another point. Move 1 unit to the right and 13 units up from the initial point. Plot this point at (1, 16).
- Connect the two points with a straight line.
Now, look for the point where the two lines intersect. This point represents the solution to the system of equations.
Based on the provided options, it seems that option D, (0, -5), is the point of intersection. However, please note that the actual solution might depend on the accuracy of the graphing and the available options.
To solve the system of equations y = 3x - 5 and y = 13x + 3 using graphical method, you need to plot the two equations on a coordinate plane and find the point where they intersect.
1. Start by plotting the first equation, y = 3x - 5.
- Choose different values of x and substitute them into the equation to find the corresponding y values.
- For example, when x = 0, y = 3(0) - 5 = -5. So, you have the point (0, -5).
- Repeat this for a few more values of x, and plot the points on the coordinate plane.
2. Now, plot the second equation, y = 13x + 3, using the same method.
- Choose different values of x and calculate the corresponding y values.
- Plot the points on the coordinate plane.
3. Look for the point of intersection between the two lines.
- The point where the lines intersect represents the solution to the system of equations.
- In this case, it looks like the point of intersection is (4, 3).
So, the answer to the question is option A: (4, 3).