Estimating a Solution via Graphs

Task

Jason and Arianna are working on solving the system of linear equations

Rounding their answer to the nearest hundredth, Jason and Arianna find that $x \approx 4.04$ and $y \approx 7.31$.

1. Explain, in terms of the slopes of the graphs of the equations in the system, why you know that there there is a unique solution to this system.
2. Show how you can tell from the graphs of the equations that Jason and Arianna must have made a mistake.
3. Give a numerical explanation in terms of the slopes and $y$-intercepts of the graphs of the equations how you could tell that Jason and Arianna must have made a mistake.

IM Commentary

The purpose of this task is to give students an opportunity use quantitative and graphical reasoning to detect an error in a solution. The equations have been chosen so that finding the exact solution requires significant calculation so that it is easy to make an error. The particular solution given comes from multiplying the first equation by 5 and the second equation by 6 but when the second equation is subtracted from the first, the $y$'s on the left hand side and the numbers on the right hand side are added instead of subtracted. Once this $y$ value is ''miscalculated'' the corresponding $x$ value comes from substituting in the second equation.

In the solution below, the graphs are produced using Desmos. Students can also sketch the graphs by hand, perhaps after putting them in slope-intercept form. Although graphing will show that Jason and Arianna are not correct, it will only give an approximate solution. The linear equations in this task are not presented in the most common format highlighting the slope and y-intercept. The teacher may wish to follow this task up with other pairs of equations in slope-intercept form, always focusing on whether or not the solution is reasonable.

Solution

1. The two equations provided are linear so their graphs will intersect in exactly one point provided they are not parallel. The slope of the line defined by the equation $6x + 17y = 100$ is $-\frac{6}{17}$ while the slope of the line defined by the equation $5x + 9y = 86$ is $-\frac{5}{9}$. Since $-\frac{6}{17} \neq -\frac{5}{9}$ these lines are not parallel and so they intersect in exactly one point.

2. The graphs of the two linear equations are shown below:

   They appear to meet when $x$ is around 17 or 18 and $y$ is about -1. This is very far from what Jason and Arianna found and so their answer is not reasonable.

3. Note that the $y$ intercept of the line defined by the equation $6x + 17y = 100$ is $\frac{100}{17}$ or about 6. The $y$-intercept of the line defined by the equation $5x + 9y = 86$ is $\frac{86}{9}$ or a little more than 9. The slope of this second line, $-\frac{5}{9}$, is less than the slope of the first, $-\frac{6}{17}$. This means that the lines will meet when $x \gt 0$. An $x$ value near 4 is far too small, however, because the $y$ intercepts differ by more than 3 and the slopes differ by a very small amount (about 0.2). So $x$ will need to be larger than 15 before the blue line drops enough, from the $y$-axis, to meet the red line.

   The coordinates of the point of intersection are $\left(18\frac{4}{31}, -\frac{16}{31}\right)$.