Question 95·Medium·Systems of Two Linear Equations in Two Variables
In the -plane, the system of equations is represented by the two bold lines shown in the figure.
If is the solution to the system, what is the value of ?
For a system shown on a graph, the fastest method is to treat the intersection of the two lines as the solution . Read that point’s coordinates directly from the grid, then do the extra computation the question asks for (here, forming the ratio ). Double-check the sign by noting whether the point is above or below the -axis.
Hints
Use what the intersection represents
The solution to the system is the point that lies on both bold lines.
Read coordinates carefully
Find the intersection point, then read its -coordinate and -coordinate from the axis tick marks.
Form the ratio
After you have , compute by dividing the -value by the -value, keeping track of the sign.
Desmos Guide
Enter points from each line
From the graph, pick two clear points on each bold line (for example, each line’s intercepts). Enter them in Desmos as points, like (0,1).
Graph each line through its two points
For the first line, compute its slope from the two points and enter the line in the form . Repeat for the second line.
Find the intersection
Click the intersection of the two graphed lines to read from Desmos.
Compute the ratio
Using the intersection values, type y/x (substituting the numbers you read) to get the value of .
Step-by-step Explanation
Identify the solution on the graph
The solution to a system graphed as two lines is the point where the lines intersect.
Read the intersection coordinates
From the graph, the two bold lines intersect at .
Compute the requested value
Compute
So the value of is .