Answer:
1st page:
To determine if Jason and Arianna made a mistake in their solution, we can examine the slopes and y-intercepts of the two equations.
The first equation, 5x - 3y = -1, can be written in slope-intercept form as y = (5/3)x + 1/3, where the slope is 5/3 and the y-intercept is 1/3.
The second equation, 3x + 2y = 7, can be written in slope-intercept form as y = (-3/2)x + 7/2, where the slope is -3/2 and the y-intercept is 7/2.
To solve the system of equations, we need to find the point of intersection of the two lines. We can see from the slopes that the lines are not parallel, so they must intersect at some point. However, the slopes are not perpendicular either, so they do not intersect at a right angle.
By graphing the two equations on the same coordinate plane, we can see that the point of intersection is (2, 2), not (4.04, 7.31) as Jason and Arianna calculated. Therefore, they must have made a mistake in their solution.
In summary, we can tell that Jason and Arianna made a mistake in their solution by examining the slopes and y-intercepts of the two equations and graphing them to find the actual point of intersection.
2nd page:
To determine if Jason and Arianna made a mistake in their solution, we can graph the two linear equations on the same coordinate plane and look for the point of intersection.
We can begin by rearranging the equations into slope-intercept form:
5x - 3y = -1 → y = (5/3)x + 1/3
3x + 2y = 7 → y = (-3/2)x + 7/2
Now we can graph the two lines. We can plot two points for each line and connect them with a straight line to obtain the graphs.
For the first equation, when x = 0, we get y = 1/3. When x = 3, we get y = 6/3 = 2. Plotting these two points and connecting them, we get:
Graph of the first equation
For the second equation, when x = 0, we get y = 7/2. When x = 2, we get y = 1/2. Plotting these two points and connecting them, we get:
Graph of the second equation
We can see from the graphs that the lines intersect at the point (2, 2), not at (4.04, 7.31) as Jason and Arianna found. Therefore, they must have made a mistake in their solution.
In summary, we can tell from the graphs of the equations that Jason and Arianna must have made a mistake because the lines do not intersect at the point they found.
3rd page:
We can determine if there is a unique solution to the system of linear equations by examining the slopes of the two equations.
The slope of the first equation, 5x - 3y = -1, can be found by rearranging the equation into slope-intercept form y = (5/3)x + 1/3. We can see that the slope of the line is positive and not equal to the slope of the second equation, which is -3/2.
Similarly, the slope of the second equation, 3x + 2y = 7, can be found by rearranging the equation into slope-intercept form y = (-3/2)x + 7/2. Again, we can see that the slope of the line is negative and not equal to the slope of the first equation, which is 5/3.
Since the slopes of the two lines are not equal, they will intersect at a unique point. In other words, there is only one solution to the system of equations.
Therefore, we can conclude that there is a unique solution to the system of linear equations given by 5x - 3y = -1 and 3x + 2y = 7, based on the slopes of the graphs of the equations in the system.
Explanation: