Mr. McKay wrote an algebra test with a total of 15 questions consisting of multiple choice and short response style questions. Multiple choice questions, x, were worth 5 points each and short response questions, y, were worth 10 points each. There were 100 total possible points on the test. Write and graph a system of linear equations for this situation to determine the number of each type of question.
10y = 100 - 5x
Cannot graph on these posts.
To write and graph a system of linear equations for this situation, let's define some variables:
Let x be the number of multiple-choice questions.
Let y be the number of short response questions.
Now, let's create equations based on the given information:
1) The total number of questions is 15: x + y = 15.
2) Each multiple-choice question is worth 5 points, and each short response question is worth 10 points. The total possible points on the test are 100: 5x + 10y = 100.
Now we have a system of two linear equations:
x + y = 15 (Equation 1)
5x + 10y = 100 (Equation 2)
To graph this system of equations, we can first rewrite Equation 1 in terms of y:
y = 15 - x (Equation 3)
Now we can create a table of values for x and y using Equations 3 and 2, and then plot these points on a graph.
Table of values:
------------------------------
| x | y |
------------------------------
| 0 | 15 |
| 5 | 10 |
| 10 | 5 |
| 15 | 0 |
------------------------------
Plotting these points on a graph, with x on the x-axis and y on the y-axis, we get a line:
| *
15 | *
| *
10 | *
|
5 |
|
0 --------------------
0 5 10 15
This line represents Equation 3, and it intersects the x-axis at x = 15 and the y-axis at y = 15.
Another way to graphically solve this system is to plot the line of Equation 1, which would intersect the y-axis at 15 and the x-axis at 15.
The point of intersection represents the solution to the system of equations, which in this case is x = 15 and y = 0. This means that there are 15 multiple-choice questions and no short response questions in this particular test.
Note: The graph can be used to find different solutions to the system of equations. For example, if we move along the line, we can find other combinations of x and y that will satisfy both equations.